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AstOF, Lrwr^X H 




Lessons in Horology 


Jules Grossmann 

Director of the Horological School, of I/Kle, Switzerland 


Hermann Grossmann 

Director of the Horological and Electro-Mechanical School, of Neuchatcl, Switaerland 


of Charleston* S. C. 
Former Pupil of the Horological School, of I#ocle, Switzerland 


The Principles of Cosmography and Mechanics Relat- 
ing to the Measurement of Time — Motive Force, 
Mainsprings, Trains, Gearings, etc. 





19TH & Brown Sts., Philadelphia, U.S.A. 

1905 . ■ 

**■ • • . 

All Rights Reserved , . - 

' M ji'^ 

HHV ] 

J ivaWe, Desr Berlin, Uermnny, oi. July 

^ 28, 1329. Ue tegan hi. 

garter in his nativB town when fllleen yesra 

,.l.i, soon moving to Berlin. He snl«e- 

quontiy -orked and studied in the Brilish 


I.le> and P»ri>, linally setlllos in Loclr, 


Switzerland, where hii great wlenllBe tri- 

umph. »ero aehieved. He has been muuh 


Locie, of whleh he became IXreelor. Hia 
.ebiBveraentB in Ihe field of horology hive 
bpeu Impoilsiit factors in the advancenient 

lln^moit profonnd knowledge of the BulJeoL 
ll waa at Ihe instancB of ihe 8«iBa Guvem- 

IrealJM "Leasons lu llorolo((y," which ii 


1 T KHM A.VN GUOSSMAKS, son ofJiilr-., 
|-1 „« ^rn i. L«lo. 8«l,..rLaud. .n 

April ^ I3S3. At Ihp ige of liitnn 

ul thB Horuidglial School of T.oele. nuder 

to Switierlaud to ulUl further [.urBmi bia 

jL^ m 

Biudies in Uigber boralogj. When uuly 

th» poaitiDii of Director of tlie Ilorol.igtcal 

■lid ElH<ro-Mei.'haalGiil Schoul, of Nuu- 

uhnlel, Swil«rl.nd. The wort of this 


HEhool 9000 became fmiiouH. beiog iwsrded 

rannj honon Bt the grent InH-rofllinoaJ ti- 

linnlioiis were oonfelred od the Director. 

" Ifflwoni in HoroloET," which complelely 

Onvnnr tlut Bilh!.-!*! In Ihanvir nnH nnifJEiM- 

'-■"'■■ 1 


5 .-. 

i-4 . 



'.I I < . 

< ■ 



« ■ ■> ' 



• « «• 


Publisher of Thb Ketstons 

• • * » • • • 

• • 

f 9 • 




■^TO one ignores the fact tliat horology has attained a highly 
prominent position among the mechanical arts during the 
past quarter of a century, anci that this lact is due to the rapid 
progress of science, industry and commerce in our epoch. If it 
sufficed formerly to know the time, approximately, and to make 
use of the indications of the sun dials, town clocks with single hands 
and other primitive instruments, modern times, with their multi- 
plied requirements, have rapidly despoiled us of this ancient 
simphcily; they demand of us an exact marking of every instant, 
'hich only modern horary instruments permit us to attain. 

The magnificent chronometers, whose superior timekeeping 
re admire to-day, are the product of Uie two-told effort; first, of 
the theorist who, by his calculations, determines all the principles; 
then of the practical workman who faithfully follows these in the 
execution of his work. 

In the period in which we live we cannot believe that we have 
yet attained the highest point of precision, but the results are 
already so brilliant that the mind now asks the question whether, 
before going farther into the technical domain, it would not be 
better first to bring to perfection the means of observation and of 
rating, which we now invariably employ with a degree of uncertainty. 
The magnificent instruments to which we have alluded are, more- 
over, still exceptions; they are very expensive as yet. So the 
most practical object of the technical study of horology is to 
approach, as nearly as possible in public timepieces, the results of 
the precision chronometer, at least so far as concerns the exactness 
of timekeeping. This purpose will surely be attained when horology, 
seconded by the admirable resources of mechanics, will entirely 
cease to be an art, too often empirical, and become a purely 
mechanical science. 

It has long been believed that the theory of horology formed a 
science by itself, independent of general mechanics, and for a long 
time the watchmaker would not listen to anything about mechanics, 


vi Preface 

pretending that it was impossible to apply its data to the minute 
pieces which compose the mechanism of a pocket watch. This 
assertion was often apparendy sustained by practical results, and 
frequendy the purely mechanical data appeared as if they could not 
be applied to horology. But this conclusion, let us hasten to say, 
was false; for the reason that the mathematical formulas employed 
in mechanics often require less development than when they are 
applied to horology. In the first case many of the terms could be 
neglected which in the second would become important. We must 
not be astonished, either, if the results are not always what we seek. 
Let lis take an example: Would one really dare to pretend that 
the laws of friction established by Coulomb, are inexact because it 
is very difficult in horology to separate friction proper from the 
influence of adhesion produced by the oil or other lubricating 
material ? This second factor, which may often be omitted in large 
mechanics, becomes, we know, an important factor in horology. 
The work which we present to the English-speaking watchmakers 
is written by watchmakers and for watchmakers, and with the idea 
that horology and mechanics are twin sisters, and that the same 
laws and the same rules control both. 

This work is the fruit of long e-xperience in the domain of 
professional instruction in horological schools. We have endeavored 
to avoid speculadons purely theoretical, as well as long descriptive 
explanations, which belong to books suited to the general public. 
If the solution of some problems cannot, in our estimation, be 
accomplished without the aid of higher mathematics, because of 
the precision required to attain the desired end, it must be noted 
that these questions can generally be put aside by those persons to 
whom the subtleties oi mathematical analysis are unfamiliar. 

It is sufficient then to recognize the fact that the calculations 
have been made to verifj' the deductions and to make use of the 
conclusions which may be drawn from them. 

We are also obliged to grade the difficulties of calculation so 
that they are presented in proportion to the development of the 
mathematical knowledge of the persons who undertake the study, 
and we follow each problem with at least one numerical application. 
When it is possible we give also together with a complicated 
solution, another similar to it, but more simple. 

plan comprises, first, a short introduction on the principles 
of cosmography and mechanics having relation to the measuring of 

Preface vii 

time. Then follow chapters which are devoted to the study of 
motive forces produced by the weight and the barrel spring, the 
calculations of trains and the theory of gearings. Then chapters 
on escapements, and finally the theory of adjusting and regulating 
forms an important part of the work, and will be treated with all 
the exactness due the subject. We will close this exposition of 
the theory by a treatise on the compensation of chronometers. 

We hope that this work will contribute its share towards 
forming a generation of capable and educated horologists who can 
assist in the development of the fascinating industry of horology. 

We owe a just tribute of appreciation to The Keystone, which 
has undertaken the publication of this work in the English language^ 
and to James Allan, Jr. , of Charleston, S. C. , former pupil of the 
Lode Horological School, who has so well performed the work of 

Jules Grossmann, 

Lode, Switaseiland. 

Hermann Grossmann, 

Neachate) Switzerland. 

^H 9 

^K 19 

^H aS 

^B 39 

H ^ 

H 37 

H ^ 

^m 39 

H 43 


enphL p 


■ 3 









I. General Principles of Cosmography Relating 
to Horology. 

Determiiiatioa of the position of a point on the terrestrial sphere 
II. General Principles of Mechanics. 


Work of a force tangent to a wheel 

General Functions of Clocks and Watches. 


Lessons in horology. 

Wheel-work. — lb purpoie in the mechaniim of dock) and watcha . 

Maintaining or Motive Forces. 

The wei^t m a motive force . , . . 

The bMTel ipring u & motive force 

a spring 43 

Measurement ol the force of a 

Thear«ticat itudy oE the momeni of a ipring** farce . 

Coefficient of elasticity 

Variation of tfie coefficient of elasticity 47 

Values of the coefficient of elasticity E 

Limit of elasticity , 

Moment of the elastic force of a spring subjected to a flexion . 

Inequality of the elastic force of the spring 56 

Length of the spring 5S 

Development of a ipring 39 

Diameter of the hub 64 

Work produced by ipring 6S 

The fuiee .... 

Calculation of the variable raiJius of the fusee's helix 6g 

. Uniformity of the force of the spring in fusee watclies 74 

Stop-work 78 

Geometrical construction of the Maltese cross stop-work : 


1J9 Purposes of wheel-work , 

Calculationj of train) 

130 Calculations of the number of turns 

133 Calculation of the number of oscillations of the balance 84 

136 Calculations of the numbers of teeth 86 

Problenii relative to the preceding qimtiaiu 8g 

Table of Contents. 3 

141-145 Numbers of turns B?-^ 

146 Numbers of oscillations of the balance 90 

147-149 Numbers of teeth 91^93 

ISO Numbers of leeth of the minute wheels. Description of this 

iM-'H Numbers of teeth of an astronomical clock 97-99 

'SS-'sS Numbers of teeth of lost mobiles loo-ioa 

'59 Indicator of the development of the spring in fusee timepieces . . loi 

i&> Simple calendar watches 104 

'61 Decitnal watches 106 

164 Calculation of numbers comprising the teeth-ranges of the wheels 

of a watch with independent second hand 109 

iS8-i59 Wheel-work of the stem-winding mechanism 118-113 

i^a Calculation of the train in a watch of the Roskopt type .... 114 


Gearings. H 

DeGnition 117 I 

Practical examination of a gearing iiS I 

Flnt — IHflance of the c«nten iig H 

'7& Primitive radii tig I 

'79 Applications lai I 

1S5 Calculation of the primitive radii 125 ' 

I93 Application of the theory of primitive radii to the escajiements . . 129 

Second — Shape of the teeth and leavei 130 

J95 General study of the transmission of force in gearings 130 

OeterminalioD of the fonni of contact in geariod* 1 36 

io9 First — Graphic method. Exterior gearing 137 

SID Interior gearing 138 

sii Second — Method of the envelopes 140 

aig Evolvents of circle gearings 146 

MO Third— Roller method 148 

121 Flank gearings 150 

136 a Determination of the profile of a tooth corresponding to a profile 

chosen arbitrarily 154 

aa6i Gearings by the evolvent of a circle 155 

Teeth-rftnge 156 

^^ Third—Total ditunet«n IS9 

4 Lessons in Horology. 

Cycloid - 13 

J3S Definition 15] 

236 Drawing oi the cycloid 15 

337 Drawing of the cycloid of a continuous movement i6( 

138 Normal and tangent to the cycloid j6! 

239 Evolute and radius of curvature oi the cycloid 16] 

140 Length of the cycloid ifil 

Cpicyclrad l£{; 

141 Definition 16^ 

lit Drawing of the epicycloid i^ 

243 Drawing of the epicycloid of a continuous movement i& 

J44 To draw a normal, then a tanget to the epicycloid ifi 

945 Evolute and radius of curvature of the epicycloid 161 

350 Length of the epicycloid i6| 

»5i-2S3 Applications 170-17: 

RelatioD of the radtui vector to the angle formed bjrtbe VBiukbl* 

raiUiii vector and the initieJ radim vector 

Table showing the angle traversed by the pinion of several ordi- 
nary gearings during the contact of a tooth of the wheel with 

the leaf oi this pinion r7j 

Calculation of the total raditn of the wheel 176 

Form of the exceu of the pinion letf in a flaak ^earin^ 176 

a68 Radius of curvature of an elipse 

270 Total radius of the pinion 

GraphicBl conitniction ef ge&ringi 

PractiCB) Bpplicstioni of the theory of gearingi 

276 The proportional compass and its use 

Tabie for using the proportional compass 195 

384 Verification of a proportional compass 196 

385 Determination of the distance of the centers of a gearing by 

means of the proportional compass and of a depthing tool . . 196 

286 The proportional compass and stem-winding wheel gearings . . . 197 

286 Gearing of the crown wheel in the ratchet wheel 198 

287 Gearing of the winding pinion in the crown teeth of the contrate 

wheel 199 

288 Gearing of the sliding pinion and of the small settiug wheel . . . 200 

289 Gearings of the dial wheels aoi 

Vaiioui calculationt relative to gearing 203 

Conical geariogi 214 

315 Form of the teeth ai6 

Table of Contents, 5 


318 Construction of conical gearings 218 

Defects which present themselves in these gearing 223 

Passive resistances in gearing 227 

326 General ideas 227 

Friction 228 

330 The two kinds of friction 229 

332 Laws of friction 229 

333 Experimental determination of the force of friction 230 

335 Table of the coefficients of friction 231 

336 Work of friction 232 

337 Angle of friction 233 

Calculation of the friction in ^earin^s 234 

339 Friction of the teeth ranges 234 

346 Friction before and after the line of centers 238 

348 1st. The wheel drives the pinion after the line of centers .... 239 

349 2d. The pinion drives the wheel before the line of centers ... 241 

350 3d. The wheel drives the pinion before the line of centers . . . 242 

351 4th. The pinion drives the wheel after the line of centers .... 244 

352 Recapitulation of the preceding calculations 244 

Calculations of the friction of pivots 245 

354 Work absorbed by the friction of the plane surface of the shoulder 

of a pivot 245 

356 Work absorbed by the friction of the cylindrical surface of a pivot 247 

357 Determination of the lateral pressure received by the pivots of the 

mobiles in a train 247 

Influence of the oil 253 

Application of the theory of gearings 255 

Functions of the heart in chronographs 255 



I. Introduction —General Priaopl«i of Cotmogntphj' Relating to Horoto^. 

1. Principles of the measurement of time. Cosmography is » 

science which has for its object the study of the different celestial 

phenomena as they are given to us by observation and calculation ; 

it comprehends also the study of the principles which are the basis 
^^pf the measuring of time. 

^^L When one proposes to measure a length, surface, volume or 
^^Height of any kind one chooses arbitrarily a unit of length, surface, 
^^Holume or weight with which one compares the object to be 
^^Eneasured, noting exactly the number of times that this unit is 
^^Hontained therein. 

^^B When the purpose is to measure the intervals of time, it is no 
^^^bnger possible to make use of an analogical method. But to 
^^^Hect this operation one is obliged to determine the space traversed 
^^Ey a body animated with a uniform or periodically uniform motion 
(31). In the first place one concludes that the intervals of time 

are proportionate to the spaces traversed by the body considered. 

2. It is necessary then to admit that all measurements of time 
must be deduced from the observation ol a regular movemenL 
Thus, formerly, one determined the fraction of timCj more or less 
great, by the running of sand in the "hour glass," or of water in 
the "clepsydra." Now time is measured in docks and in watches 
by the periodically uniform movement of the pendulum or balance 

3. Units of time. Sidereal day. Solar day. For the determi- 
nation of the unit of time it is necessary to choose the most unifonn 
movement possible, a movement whose speed must be the same 
to-day, to-morrow, in a year or in an indefinitely prolonged period. 
Such a movement filling this condition absolutely, is the rotation of 
the earth on its axis ; no cause or effect whatever could increase or 
diminish it. We have positive proofs that this movement is the 
same to-day as it was in the time of Hipparchus, an ancient 
astronomer of the school of Alexandria who lived two centuries 

lo Lessons in Horology. 

before Jesus Christ. We can assure ourselves by the calculation 
of the eclipses, that the length of one of these rotations is the same 
to-day as in the time ol that astronomer within jj-j of a second. 
This movement has then been chosen because of its great regu- 
larity as the basis for the measuring of time. 

The duration oi a complete rotation is the unit, and is called 

4. In order to determine with exactitude the commencement 
and the end of this movement, it is necessary to choose a point of 
repose outside ol the earth, and for this purpose a fixed star or the 
sun is taken. Let us remark that the result differs according as we 
take one or the other of these two points. The following demons- 
tration will explain the reason. 

5. We know that the earth not only turns on its axis, but that 
it has also a simultaneous movement around the sun. Let us take, 
then T and T' . Fig. i, as the two positions which the earth 
occupies in its orbit at the commencement and at the end of one of 
its diurnal rotations. In the first of these positions, a is a point on 
ita Hurface Irom which can be seen at this instant the center of the 
■uii S, in an imaginary plane passing through the two poles and the 
point considered a, this plane is the meridian plane. At the end 
of a certain time, the earth has traveled in its orbit to the position 
T* and the point a arrives at a' in such a manner that the line 
7* a' is parallel to T a. The earth wHl then have accomplished 
one rotation on its axis and all of its parts will have, with relation 
to the fixed stars, the same positions that they had at T. 

The time during which this rotation is accomplished is called^ 
H lidereal day, 

But from the point a' in the position T' one could not see the 
•Un in the meridian plane ; in order that the observer placed at t£ 
could perceive it anew in this plane, it would be necessary for the 
pi)lnt «' to be removed to b in traversing the arc a' b. The solar 
ttt^y, thut is to Biiy, the time which elapses between two consecutive. 
IWHHUgeN (i( the nun to the meridian plane is then longer than the 
■IflvrcHl (tiiy. In ilividing the solar day into 24 hours, the hour 
Intrt ficj iTihiiit<-N imd the minutes into 60 seconds, the sidereal day 
uniiiitH only -J3 h(jun<, ,<^6 minutes, 4.09 seconds ; the sidereal day is 
Iht^ii ■Imrtrr thun the hoW day 3 minutes. 55.91 seconds. 

It we divide, on the other hand, the sidereal day into 24 hours, 
Ih" ttulm diiv will count 34 hours, 3 minutes, 56,55 seconds. Thia 

General Principles of Cosmography, 


value of the solar day, variable from the sidereal day, is only a 
mean value. 

6. The time that the earth takes to traverse its orbit, that is to 
say a year, contains exactly one sidereal day more than the solar days. 

7. True time. Mean time. The curve that the earth describes 

Fig. 1 

around the sun is an ellipse of which that star occupies one of the 
foci. Our planet does not traverse this curve with a uniform speed, 
it moves more rapidly when it is nearest the sun and more slowly 
when it is farthest away. The arcs traversed by the earth in one 
day are not then the same length during all the year. There 
results an irregularity in the duration of the solar day, the solar 
day is longest when the earth goes fastest, and it is shortest when 

Lessons in Horology. 

its movement of translation is slowest. Another cause which still 
increases this irregularity is due to the fact that the axis of the earth 
is not perpendicular to the plane of the orbit that it traverses around 
the sun (plane of the ecliptic). 

The length of the solar day can vary in 34. hours as much as 
30 seconds, plus or minus. Thus the solar day with its diurnal 
variation of length does not fill at all the conditions desired for the 
measuring of time, the unit adopted must be of fixed value, so that 
our horological instruments, all based on a uniform movement, 
may follow their regular running without necessitating perpetual 
resetting. We fall then into a difficulty, since naturally the sun 
should measure the time for us, while in reality its unequal move- 
ment does not lend itself to this measuring. The difficulty has 
been adjusted as follows. We divide the total duration of the year 
by the number of solar days that it contains ; the quotient will 
be a mean value, shorter than the solar days of greatest length and 
longer than the solar days of least duration. It will be, moreover, 
almost equal to certain days between them. This mean value is 
called ?nean lime. We call, on the contrary, Inie time the direct 
interval of time elapsing between two successive passages of the 
sun across the meridian. The difference, plus or minus, between 
true time and mean time can amount to as much as 17 minutes. 

8. The equation of time is the value that must be added to or 
subtracted from the true solar day to obtain the mean solar day. 
The year book of the Bureau of Longitudes announces each year in 
a calendar the result of the equation of time, and gives in a 
column entitled "Mean time at true noon" what a chronometer, 
regulated on mean time, should indicate at the exact moment 
of noon. The equation of time is nothing or almost that, four 
times a year — the 15th of April, 15th of June, 31st of August and 
25th of December, while it attains its greatest value between the 
loth and i2th of February and the first days of November. 

9. Laylnff out of a meridian line. We already have an idea of 
the importance of the meridian plane in the determination of the 
length o! a rotation of the earth on its axis. 

Let us see now how we can proceed to establish the direction 
of a meridian line, that is to say, of the trace of the meridian plane 
on the surface of the earth. Among the several methods known let us 
choose the following, which recommends itself on account of its ex- 
treme simplicity, and which does not require instruments of precision. 

General Principles of Cosmography. 13 

On a horizontal plane, conveniently placed, we fix a vertical 
style ; from its toot, O, Fig, 2, we describe on this plane several 
concentric circles of any size, such as m n, m'n', etc Let us mark 

ton these circumferences the points A, B, C, etc. , where the extremity 
of the shadow of the pin reaches Ijefore midday. In the afternoon 
renew the operation by indicating in the same manner the points 
, if, C, etc. We connect the points marked on the same cir- 
cumference by a straight line, and we obtain thus as many straight 
lines as circles, and they are parallel to each other. The perpen- 
dicular laid o5 from the center O, on these straight lines, will be 

the meridian line sought. Since the shadow cast by such a style 
is never very distinct, we will reach a greater accuracy by finishing 
the extremity oi the pin with a metallic plate, in which we pierce a 
fine hole. We indicate then the center of the image of the sun on 
each of the circumferences, as previously done. In the above con- 
struction a single arc sufficed, but it is preferable to employ several 
hich mutually control each other, the middle of the straight lines, 
'A', BB', CC, etc., should be found with the center O on the 
le perpendicular N. 
10. When the middle of the small image of the sun is found on 
the meridian it is nearest the foot of the style and consequendv the 
sun is at its greatest height ; it is then exactly midday. To obtain 
the mean hour we must consult a table ol equations and add or 

14 Lessons hi Horology. 

subtract, according to the season of the year, the correction indi- 
cated for this day. Generally, as we have said, these tables indicate 
the "mean time at noon," thus for the 17th of November, 1893, 
we find for example indicated : 

Mean time at noon 1 1 h. 45 m. 10 s. the clock or watch should 
then mark 11 hours, 45 minutes and 10 seconds when the middle of 
the small image of the sun is projected on the meridian line, 

U. The Meridian Glass employed in the observatories is nothing 
else than a meridian line determined with the greatest exactness. 
It is generally a glass of sufficiently large size which can only move, 
in the meridian plane, and is divided by it into two symmetrical 
parts. It is supported on two immovable pillars by means of trun- 
nions, which permit it to take all the positions possible around its 
axis of rotation. It can then be used to observe the passage on the 
meridian of all the visible stars above the horizon. The sensii 
ness of the instrument is moreover augmented by the magnifying 
power of the glasses employed. 

Since such an instrument cannot be transported, we have 
recourse to other instruments in order to determine the hour at any 
locality, on the sea for example. The one generally used is the 
sextant, but its employment is complicated. 

12. If the mean hour is known, and it is only a question of 
maintaining it, the apparent motion of the fixed stars is easily used 
for this purpose ; in short, since these stars return to the same 
position at the end of 34 sidereal hours, it is sufficient to place j 
level of any sort, but fixed and invariable, in the direction of a star 
the next day at the same hour (solar time), less 3 minutes, 55.9: 
seconds this same star would present itself anew before the level 
The fixed stars afford the greatest facility for the control of the 
running of watches and clocks. 

13. Determination of the position of a point on the terrestrial 

sphere. Since marine chronometers are among the instruments 
wiiich are used to determine the position of any point on the surface 
of the earth, especially that of a vessel at sea, each watchmaker 
should inform himself of the part that these instruments play in 
■uch observations, on which depend the security of the ship and 
that of the beings which it transports. 

14. In ordtT to represent the position of a point on the surface 
(if a sphere, such as the earth, we suppose described on this 
sphere two great circles, one, passing through the two poles, 

General Principles of Cosmography. 15 

called the meridian circle; the Other, perpendicular to the 
first, is the equator. This last is consequently perpendicular 
to the terrestrial axis, and at all points equally distant from the 
poles. Each of these circles is divided into 360 degrees. The 
divisions marked on the meridian circle commence at the equator, 
and are reckoned north and south to the poles, therefore from 
o to 90 degrees. These degrees are called degrees of latitude. 

Since we can imagine an infinite number of meridians passing 
through the poles, the point o can be placed arbitrarily, that is to 
say, the first meridian at whatever place it suits best. Thus 
England has chosen as the starting point the meridian which passes 
through the Observatory at Greenwich, in the neighborhood of 
London ; France has made choice of the one which passes through 
the Observatory of Paris, and other nations have made their first 
meridian pass through the Isle of Fer. 

The degrees reckoned on the equator are called degrees of 
longitude, and are reckoned both to the east and to the west of the 
first meridian, from o to 180 degrees. 

By imagining circles parallel to the equator passing through 
each division of the meridian circle, and meridian circles passing 
through each division of the circle of the equator, the latitude of a 
point will be then the distance in degrees from the parallel circle 
passing through this place to the equator, and its longitude will be 
the distance in degress from the meridian of this place to the 

idian chosen as the starting point. 

These values constitute what are called the geographical 
'<rdinafes of a point, and the position of this point on the terres- 

globe will be perfectly determined when we know its longitude 
east or west, and its latitude north or south. Thus we would say 
that the geographical coordinates of the city of Neuchatel, in 
Switzerland, are 

^H 4G° 59' 15" north latitude, 

^p 4° 35' 54" longitude, east of the meridian of Paris. 

15. In order to determine practically the latitude of a point A, 
Fig. 3, the simplest manner is to measure the angle formed by a 
horizontal line A B, and by the line A C ending in the Polar star. 
In short, the fixed stars being prodigiously removed from the 
earth, which is but a point in relation to this enormous distance, 
we can say without appreciable error that all the straight lines 



Lessons in Horology. 

drawn from the earth to the Polar star are parallel to each other, 
which moreover conforms to experience. 

Since the Polar star is found almost exactly on the prolongation 
of the axis of the earth, the straight line that we imagine drawn 
from any point on the globe to this star is parallel to the axis. 
We can then say that the latitude of such a place as A, which is in 
reality the angle A O E, is represented by the angle B A C; 

practically the angles A O E, and S A Care equal by having their 
Ndes perpendicular each to each. We see then that the latitude of J 
a place is equal to the height of the pole above the horizon. In J 
order to determine it with greater exactness it would still be neces- 
sary to take note of the distance of the Polar star from the axis and ' 
of the refraction of the luminous rays. 

16. The means employed in order to determine the longitude 
by direct observations are scarcely practicable, therefore we use in 
preference a marine chronometer whose daily rate is known. 

Let us suppose that a ship is one day at a certain point, which 
we will designate by A, that we have made the observation of the , 

General Principles of Cosmography, 17 

passage of the sun across the meridian, and that we have noted the 
advance or the delay of the hour marked by the chronometer at 
this instant. Let us call Q the number of hours, minutes and 
seconds shown by the watch at the exact moment of the passage of 
the sun across the meridian. One or several days later, we repeat 
this observation and we find, let us suppose, a new correction that 
we will designate by C. 

The difference between Q and C, divided by four minutes, 
will give us the number of degrees that the ship has advanced in 
one or the other direction of longitude. 

In short, since the earth takes twenty-four hours to accomplish 
a complete rotation of 360 degrees, it would require ^^ which is 
four minutes to cover one degree. We understand that it would 
still be necessary to take account in the calculation of the daily rate 
of the chronometer ; this value is an average based on a great 
number of former observations. All the accuracy of this method 
depends then on the absolute regularity of the daily rate of the 
marine timepieces. 

The values C^ and C are algebraic values, that is to say of 
quantities preceded by positive signs ( + ) or negative signs {—), 
since the chronometer can be behind or in advance at the time of 
the passage of the sun across the meridian. 

Let us determine now which of these signs belongs to the 
advance of the timepiece and which will designate its retardation. 

In the first place, it is evident that this choice is arbitrary ; 
thus, at the Observatory of Greenwich, they have adopted the 
negative sign for the retardation and the positive sign for the 
advance. This choice seems at first sufficiently normal, but at 
Neuchatel one is placed at a different point oi view. When the 
chronometer is slow at the passage of the sun, in order to find the 
true time we must add to the hour indicated by it the correction 
C; this value should then be preceded by the positive sign, while 
when the chronometer is fast at the passage of the sun we must 
subtract the correction C in order to obtain the correct time ; in 
this case then, this value should receive the negative sign. It is 
this manner oi viewing the matter which has led, at the Observatory 
of Neuchatel, to the adoption of the sign -f fc slow, and — for 
the advance of the chronometer 

\X. As we have said that a chronometer never follows exactly 

mean time, its daily rate therefore should be determined in an 

1 8 Lessons in Horology. 

observatory : because this daily rate must be reckoned with in the 
determination of the longitudes. Let us designate by A this mean, 
value, and since it can be fast or slow let us precede it by the 
negative or positive sign. The chronometer la observed at the 
moment of the passage of the sun across the meridian ; this obser- 
vation giving the true lime it will be necessary to deduct the mean 
time from it Let us call B the diSerence between the mean time 
and the true time (equation of lime) and let us determine the sign 
of this last value. 

Since it is desired to bring back the correction C to the mean 
time we will argue that, if the mean time at noon is fast, the true 
time is slower than the mean time and the value B should receive the 
sign +, on the other hand, if the mean time is slower than the true 
time, B would receive the sign — . 

Let suppose now that a ship leaves a seaport whose longitude 
is E^ degrees west of Paris. The day of departure we have ob- 
served the passage of the sun and obtained a correction Cq . The 
correction C'^ of the chronometer on the mean time will be for this 

C\ = C„ + B^ 

at the end of A^days we repeat the observation of the passage trf- 
the sun, and we will obtain a correction C between the time of th^. 
chronometer and the true time : the correction C between the' 
time of the chronometer and the mean time will be expressed by 

C' = C+ B. 

The difierence D betn-een the time of departure and the time 
of the place where the vessel now is will be 

Z> = ( r+ J) — (C„ + J„) - NA. 

Reducing this value to minutes, we will have the longitude B 
in degrees by the division r 

18. Let us take a numerical example. 

The longitude of Havre west of the meridian of Paris being;! 
a" 13' 45". let us imagine a vessel leaving this port November 2, 1 
1893. The time shown this day by a marine chronometer at the 
moment of the passage of the sun across the meridian is 11 h. 

General Principles of Mechanics. 

4J m. 42 s. The equation of time for this date is + 16 m, 21 s. 
We will then have noted the correction C'^ — 11 h. 43 rn. 42 s. 
+ 16 m. 21 s. = 12 h. o m. 3 s. Four days after, a new 
observation shows that at the moment of the passage of the sun 
the chronometer indicates iih. 28m. 57s. For the 6th of Novem- 
ber, the equation of time being + i6m. 15s, the new correction 
will be C= II h. 28m. 57s. + 16m. 15s. = II h. 45m. 12s. 
The diSerence 


ZJ- i3h. o m. 3 s. — iih. 45m. \i%. — NA 

I subtracting 

D=i\\a. 51 s. - NA. 

Supposing that the mean daily rate of the chronometer be A 
0.5 s, we will have 

We will have then the longitude sought by 

y performing the division we will obtain 

£ = 3° 4a' 13" + 3° ij' 45" = 5° 56'. 

The ship will then be at noon on the sixth of November 
gitude 5° 56' west of Paris. 

II. General Prindplei of Mechuiici. 
Forces. Any cause which produces or modifies the move- 
ment of a body is a. force. A force can hs power or resistance, that 
is to say, it can, without losing its active character, act in the same 
manner as or contrary to the movement. Such are the effects pro- 
duced by animated beings, by wind, steam, waterfalls, etc. 

Passive forces exist naturally and can partially or totally 
destroy motion, but are incapable of producing it; such are, among 
others, the effects produced by friction, the resistance of the air, etc. 
20. We can estimate very accurately the greatness of forces 
by their eflects. The value of a force can always be represented 
by a weight, as the kilogramme or gramme, which would make 
equilibrium with it. Thus the force exerted by a man in order to 
^Hlt a car in motion can have its greatness measured by a certain 

20 Lessons in Horology. 

number of kilogrammes. Let us suppose, for instance, a cord 
fastened to the car and passing over a fixed pulley placed before 
the vehicle ; it we suspend weights to the free end of the cord and 
increase them until the car commences to move, the total oi the , 
weight will give us the measure of the eflort put forth by the maaiS 
in order to produce the movement desired. I 

21. As a general rule, we give the name of violive force to any ' 
power which puts a body in motion, and, on the other hand, that 
of resistant force to every active or passive force in opposition to 
this movement. 

22. Without being able to define the nature of forces, thfr 
sensations which they invariably produce in us give us immediately 
an idea of their intensity and of their direction. 

The directions of forces are represented by the straight lines 
along which they tend to move the body to which they are applied. 
It is suitable to represent their intensity by lengths which are pro- 
portionate, the result is that we can submit forces to the ! 
mathematical processes as any other quantities. 

23. The point of application of a force is that part of a body 
on which it acts direcdy in order to change the state of motion or, 
of rest of this body. 

24. The line of direction of a force is that along which it tend* 
to make its point of application advance. 

25. A force capable of replacing by itself alone a system cj 
forces acting on a body is called the resultant of all these forc**s' 
These last are called the components of the only force able t 
replace them. 

26. The trajectory is a line which the movable point follows^ 
The movement is called rectilinear or curvilinear, according as tl 
trajectory is a straight line or a curve. 

27. Law of inertia. Experience has established a law 1 
which all bodies are subject and which constitutes a fundamental 
principle of mechanics. This law known under the name of "prin^ 
cipie of inertia'"'' can be defined as follows : 

A material body cannot put itself in motion if it is at rest and,' 
reciprocally, if it is in motion it cannot of itself modify its movement, 

28. Definition of mechanics. Mechanics is the science of forces 
and their effects. Its object is to find the relations of the forces 
which affect a body or a system of bodies causing this body or this 
system to take a certain movement in space. Reciprocally beingj 

General Principles of Meckania. ai 

given a body or a system of bodies, to find the motion that this 
body or system of bodies will take in space under the action of 
given forces. 

This general problem comprehends the one in which the forces 
make no change in the state of the body or of the system, a 
particular case in which we say that the forces are in equilibrium. 
Thence comes the division of mechanics into sialics or the science 
of equilibrium and dynamics or tlie science of motion. We can 
still study the movements of bodies, considering only their direc- 
tion, intensity and duration, in leaving out the matter of which the 
bodies are formed and the forces which produce or modify these 
movements. This study forms a part of mechanics to which is given 
the name of kinevialics, which can also be called geometric mechanics. 

29. notion. Motion is uniform, when equal distances are 
traversed in equal times. 

We call the space traversed in the unit of time velocity, we 
will have then, designating velocity by v, space by s and time by /, 

whence we have 

\ = \ and t = - 

30. The motion is called variable if the spaces traversed in 
any equal times are unequal ; that is to say, when the speed of the 
ixidy is not constant during the entire duration of the motion. 

31. When a moving body traverses certain equal distances In 
equal times, without fulfilling the same conditions for parts of these 
distances, we say that the motion is periodically uniform. Such 
are, for example, the motion of the earth around the sun and the 
vibratory motion of a pendulum in small amplitudes. 

32. The motion is uniformly variable when the velocity of the 
moving body varies equal quantities in equal times. 

The acceleration is then the quantity which the velocity varies 

ig the unit of time. 

If in uniformly variable motion, the velocity increases the 
acceleration is positive and we say that the motion is uniformly 

If the velocity diminishes, the acceleration is then negative 
and the motion is said to be uniformly retarded. 

33. The motion of a body which falls by the action of its 
weight is uniformly accelerated. In this case we designate the 

leration due to the weight by the letter g; this value is constant 



a Lessons in Horology. 

for the same place and in our regions g = 9.8088 m. This v^uj 
represents twice the distance traversed during the first second bjfl 
a body falling freely and without initial velocity, 

34. Rotary motion. A solid is animated with a movement of 
rotation on an axis when each of its points describes a circum- 
ference whose plane is perpendicular to the axis and whose center 
is found on this axis. In this movement any two points of the body, 
describe, in the same time, similar arcs, that is to say, of the sanM 
number of degrees ; but the lengths of these arcs are diflerenl 
and should be proportionate to their distances from the axis. Let 
e and e' be the arcs traversed in the same time by two points » 
and m' (Fig. 4) situated at the distances r and r' from the axis O 
rotation, we would have 

The movement of rotation is uniform if, in equal times, a poiq 
of the body describes always equal arcs. 

The velocity of such a raotitw 
can only be determined by con^ 
sidering at the same time th6 
path traversed in i second by i 
point of the body and the distance 
of this point from the axis of rota- 
tion. In order to avoid this double 
data, we consider the points whidt 
are at the unit of distance from 
the axis and we call the angular 
velocity the length of the arc 
scribed in i second by a point 
ng. 4 situated at the unit of distan< 

from the axis. 

Let w be this arc, we will have, for the velocity v of another 
point situated at the distance r 

from whence we conclude w ^= ^- Consequently, the angulai 
velocity is obtained by dividing the space which is traversed \ 
any point in i second, by the radius of the circumference which 
it describes. 

35. Hass of a body. The quotient of the weight of a fjody 
in any place on the globe, by the acceleration due to gravity at this 

General Principles of Mechanics. 23 

place is constant. This value is what is called in mechanics the 
mass of the body. This quantity, which is of a particular nature 
since it is nothing else than a quotient, can be subjected to calcula- 
Q just as any other quantity. In designating it by M, we have 

^^H M ^ — from whence Mg = iV and g — ^ 

^^H 36. The product Mv of the mass Moi a moving body by the 
^^■kxuty which it possesses, takes the name of quantify of motion. 

H ^ 

^^piict of the intensity of this force by the path traversed by its point 
of application. In other words : 

The work produced by a force constant in magnitude and in 
direction is represented by the product of the intensity of this force 
by the projection of the distance traversed by the point of applica- 
tion on the direction of the force. 

*Thus, the path and the force being in the same direction, we 
have (Fig. 5) 

Work of a Force. 
37. Deflnition. We call in mechanics work of a force the pro- 

^ FY. A B. 
If the path A B and the force F have not the same direction, 

we will project in this case the path A B q\ 
ewill have (Fig. 6) 

W^ FY, A C 

the direction A F and 

Let us observe that the projection of the path on the force is 
IS the angle B A C becomes smaller ; the work will then 

I greatest when A B and A F will have the same direction, 
! angle B A F becomes greater, the projection diminishes, and 

and ^ 

38. Work of a force tangent to a wheeL On imagining the 

movement of rotation of the wheel to be very small, we can admit 
that the movement takes place along the tangent Let us call 
F the force and s the space traversed in its direction, we wDl have 
for this slight displacement 

W = FY^s. 
When the wheel will have made a complete revolution, the 
path s will have become a circumference and we will have the work 
for one revolution expressed by 

W^ Fy. j^r. 

39. Unit Of work. We have chosen lor uriil of work that 
which is the product of the unit of force by the unit of distance, 
that is to say, in meclianics, of the kilogramme by the meter and 
in horology, of the gramme by the millimeter. We have given to 
this unit the name of kilogrammeier for the machines and of 
grammilli meter for the more delicate pieces of horology. As 
abbreviation, we will designate by the letters gr. m. this last unit 
which we will make use of in the entire extent of this course. If, 
for example, the force is 3.5 grammes and the distance 0.4 milli- 
meters, the work of the force will be 

B^= 3.5 X 0.4 = M gT. m. 

40. Active power. A weight P which falls from a height h 
generates a certain work that we can represent by the product 

We find in mechanics that every body falling from a height k 
is animated by a velocity v which is connected with the height k by 
the relation i/> = 3 gh, 

from which expression we can draw 

'•- f,- 

Replacing in the equation of work Ph, h by this last value, we will 

but — being the mass w of the body, we will have at length 

General Principles of Mechanics, 


The expression i m v^ has received the name of active power. 

We can then say that the active power of a body in motion is half 

the product of its mass by the square of its velocity ; or, also, that 

the mechanical work which imparts a certain velocity to a body is 

equal to the active power which animates that body. 

We give the name of active force to twice the active power ; 

we have then 

Active power = \ tnv'^. 

Active force 

tn v^. 

4L Every body in motion is capable of doing work. In effect 
the body has a mass m ; it has a velocity v, since it is in motion, con- 
sequentiy the product ^ m v^ gives us the value of the work Pky 
to which the velocity v corresponds ; we can therefore say that 
every body in motion is capable of producing work. 

Moment of a Force. 

42. Let us now imagine two cylinders of different diameters 
turning around an axis O (Fig. 7) and let us admit, for example, 
that the first is three times as great as the second. Let us 

suspend weights at the ends of light cords 
wrapped around each cylinder, in such a 
manner that each of these weights acts in a 
contrary sense to the other. In order that 
equilibrium may exist in this system, we will 
find that the weight fixed to the small cyl- 
inder should be three times as great as 
that which is fixed to the large one. In 
this manner, if we turn the cylinder one 
revolution, one of the weights will rise 
^ while the other will fall ; the weight p tra- 

il P LiP versing a path represented by 2 r X 3 its 

^^' "^ work will be 

The weight P traverses a path 2 tt X i, producing at the same 
time a work 

Since we have 3 / = P we can admit 

(a) 2 TT X 3 / = 2 'T X I Z', 

and there will be equilibrium, because the mechanical work of one 
of the weights is equal to that of the other. The equality of the 


Lessons in Horology. 

work of these two weights will exist also when we make the cylin* 
ders describe only a fraction af a revolution : as small as this 
fraction may be. 

Dividing the equation (a) by 2 tt we obtain 
3 / = I P. 

The figures 3 and i are the respective lengths of the radii of each 
cylinder ; this radius takes the name of lever arm and the abovtf 
product of the intensity of the force by the lever arm is called thd 
moment 0/ ike force. 

Summing up, we have just examined the state of a body 
which can turn around a fixed point ; to this body are applied two 
forces whose work mutually counteracting, produces equilibrium. 
We call such a system a lever. 

In every lever, lor equilibrium to exist it is therefore necessafjF 
for the moments of the two forces in action to be equal. 

Imagining the system in motion, under the action of an exterio* 
impulse, we will find the work of the forces on multiplying theif 
moment by the angle traversed ; in this new condition of the body 
the work of the forces will then be equal. 

43. Lever. Pracdcally, a lever is a solid body, movaUfi 
around a fixed point and acted upon by two forces tending t<I 



n*. 8 

make it turn in contrary directions. Fig. 8 represents a '. 
in which O is the fixed point. P and F the two forces, 
lever arms of the forces P and F are the distances from the fixt 
point O to the two forces, that is to say, the perpendicular O A 
and O ff dropped on the direction of these forces. 

General Principles of Mechanics, 27 

From what we have said before, equilibrium will exist when 
the moment of the force P will be equal to the moment of the 
force Fy that is to say, when 

which can be written 


The two forces should, therefore, be in inverse proportion to their 
lever arms. 

Fisr. 9 

44. One often distinguishes two kinds of levers : In the lever 
of the first kind the fulcrum is situated between the points of 
application of the two forces ; in the other, Fig. 9, this fixed point 
is situated at one of the extremeties of the body. From the 
theoretical point of view this distinction is useless and the con- 
ditions of equilibrium of the lever apply to all cases. 

Transmission of Work in Machines 

45. We give the name of machine to every system of bodies 
intended to transmit the work of forces. In order to explain in 
what manner this transmission is effected it is necessary to enter 
into some details. 

The relative movements of the different parts of a machine are 
not determined only in direction but also in intensity. Generally 
the movements are periodically uniform (31) ; the speed is put in 
harmony with the requirements of the industrial work to be 
produced without its ever attaining the limit at which the solidity 
of the machine would be endangered. 

A6* Different forces act on a machine in motion, which can be 
divided into three classes : 

1st Motive forces. These are those which act in the direction 

98 Lessons in Horology. ^H 

of the movement of the parts which they operate ; it is consequent^ 
Uj these that is due the motion of the machine. 

2d. Useful resisting forces, those which the materials on which 
the machine operates, oppose to the movement of the parts which 
act on lliem ; it is these then which we desire to overcome. 

3(1. Passive or hurtful resisting forces which arise from the 
movement of the ditlerent parts of the machine to oppose this 
movement. We have already seen that they are due to the friction 
of these parts among themselves or on foreign bodies, to shocks 
which can be produced between these parts on account of sudden 
changes in speed and to the resistance of the air. 

CunBidering the motive forces as positive, since they act in the: 
direction of the movement, the useful resistances and passive 
rCBiHtanccM will then be negative. If we suppose the system 
KUimatcd by a uniform movement, the algebraic sum of the work of 
ull the forces for any given time will be null, since the gain or the 
loM of active power is null, and we will have, in designating by 
Wn the work of the motive forces, W» the useful work and Wp the 
work of the passive forces : 

J/-. -W„ —W), = o, 
Ironi whence 

W,» --= HI + Wp, 

which shows lis that, the movement being uniform, the motive worto 
1b equal to the useful work, augmented by the work of the passive 

When in any machine this formula is verified, we say then that 
there is "dynamic equilibrium," 

When the movement of a machine is periodically uniform, the 
gnin or the loss of active power is null only for a whole number ol 
[>eriod» : for this time we still have 

Wm -- IK. + n'p 

Wc say then that the machine is in " periodic dynamic equi- 
llbrlnm;" lhi» is the ordinary state of machines, not only on 
Aect^Hlll o( ihc aliapc ol their |n»rts, but because of the variations: 
miMT or Inw grrttl in the iuoii\-e fortes, iind espedally in the; 

Thtw M', U nlwnj-s inferior to »*« ; that is to say, a machine- 
remlont |m* iwHiil work than the motive power api^ied, because 
ihc tt\uk ol iho [>ttWLi\v rciiat*«ices is ncwr null 

General Principles of Mechanics. 29 

At. Calling P the motive force acting on any machine, and Q 
the useiul resistance overcome by this machine, E and c being the 
spaces traversed by the points of application oi P and of Q in the 
direction of these forces and in any equal time, at the beginning 
and at the end oi which the speed of the machine is the same, the 
equation of dynamic equilibrium gives, supposing first that the pas- 
sive resistances are null : 

PE^ Qe. or^ -- | 

From the equality between the work and the power and that 
of the resistance, it follows that for the same motive work P E, 
according as the force ^ may be multiplied by J^, )4, 2, 3, etc., 
the space e will be divided respectively by the same numbers ; from 
whence comes the maxim well known in mechanics: "That which 
we gain in force we lose in speed, or what amounts to the same 
thing, in distance, and reciprocally." 

The preceding proportion enables us to calculate any one of 
the four quantities P, Q, E, e, when we know the three others. 

For any simple or complicated machine, il the question is to 
find the resistance Q that a power P can overcome, we determine 
the spaces E and e traversed in the same time by the points of 
application of the forces P and Q. E and e are any distance 
whatever if these points of application have uniform movements, 
but we take them corresponding to a period if the movement of the 
machine is periodic. When the machine is constructed, by putting 
it in motion of any sort, we determine the values of E ; we deduce 
those of e from the relations of the spaces traversed by the different 
parts which transmit the movement of the point of application from 
P to that of Q. 

Let us suppose that the resistance to be overcome Q be 100 
kilogrammes, and that it is desired to determine what will be the 
power of P. neglecting the passive resistances. We commence by 
determining the corresponding values of E and e, as has just been 
shown. Let £^ 2.5 m., and ^^o.Som. ; replacing the letters by 
their values, in the preceding equation we will have 


P= — ^j ' ^ 32 kilogrammes. 
It we had known P we would have been able to determine Q, as 
have just done tor P, 

Lessons in Horology. 

48. In machines, especially in industrial machines, the passive 
resistances are so considerable ihat we cannot neglect the work that 
tfacy absorb ; the dynamic equilibrium is then expressed by 

W„ = IVu + IVp . 
For a certain displacement of the parts of the machine, the 
work Mffl IVu Wp will be valued as in the preceding case ; thus 

/* being the power, Q the useful resistance, R, R' the difTerent 

{tassive resistances, and E, e, i, i" . , , . the corresponding distances 
traversed in the same time by the points of application in the 
direction of these forces, we have 

PE^ Qe + Ri-^ R'i' ^.... 

49. It may happen that one or several hurtful resistances come 
from the shocks between the parts of the machine. The work 
absorbed by these resistances is no longer valued by the product 
of a force by the distance that its point of application traverses, but 
by the loss ol active power due to the shock, and this loss, valued 
in units of work, enters into the second member of the equation as 
the other hurtful works R i R" i'. . . . 

By the aid of the preceding equation, knowing in a machine 
two of the three following works : the Wn ^ P E, the Wi =Q e 
and the W^= R i -^ R' V ^ we determine the third. 

50. Ordinarily one decides to set up a machine capable of 
producing a given useful work. 

WH = Qe 

It is then necessary to determine the Wm =^P E, capable d 
producing not only this useful work but of overcoming also the 
secondary resisting works. 

One should then commence by calculating this hurtful work, 
which is done by determining the values of the diSerent passive 
resistances R, R" . . . . in function of Q, and afterward Wp in 
function of H^« . 

Having Wp and IV^ , we can determine the value of Wm 
expressed us has been said in kilogrammetere and in grammilli- 

5L The motive work Wm being represented by loo, the useful 
and hunful works H^ and Wp being for example. 75 and 25, the 
loM is then 25 for 100; we say in this case that the product of the 
nmchinc is 75 per cent. If it were possible for the loss to be 
nothing, the product would be 100 per cent ; this fact can never be 

General Principles of Mechanics, 31 

realized, which renders absolutely illusive the hypothesis of perpetual 
motion. The product of a machine rarely passes 80 per cent. ; it 
is nearly always much inferior to this limit. 

In this preliminary study we have desired to establish a basis 
which is nothing more than the enunciation of some fundamental 
principles of mechanics. In the course which is about to follow, we 
will make a constant use of them, and all their developments will 
be found in the text. 


General Functions of Clocki and Watches. 

The OiollatioDs of the Pendulum and th^ Relation to the Motive Force. 

52. We know that in clocks and watches time is measured by 
the periodically- uniform movement of the pendulum or of the 
balance wheel. 

History relates that Galileo, while yet young, was struck with 
the regularity of the pendulous vibrations of a candelabra in 
Cathedral of Pisa. He studied the laws of these oscillations and 
used a pendulum later on for his astronomical observations. This 
instrument, in its primitive simplicity, presented two difSculti 
when the astronomer left his pendulum to itself, after having 
diverted it from the vertical position, the oscillations which ■ 
produced having at first a certain amplitude, diminished litde by 
little, then finally ceased entirely. He was then obliged, from time 
to time, to give an impulse to his pendulum. The second of these 
difBculties was the necessity for him to count the number of these 
oscillations. It is said that he charged a servant with the execution 
of these two functions. 

Now, the mechanism of the clock performs unaided these two 
functions with a regularity that the man could never achieve direcdy, 

53. Let us seek, in the first place, for the causes which make 
the oscillations of a free pendulum constandy diminish. 

When a pendulum is moved from its position of equilibrium 
O A (Fig. lo), the attraction of the earth, which was perfectly 
neutrahzed by the resistance of the point of suspension O, is no 
longer so in the oblique position O B. It would cause the ball to 
descend vertically if the cord did not force it to describe the 
arc of a circle; at each instant of its returning course the speed 
of the pendulum increases a small quantity until it reaches e 
the vertical position O A. From there on the inertia, or, if one 
prefers it, the velocity acquired, forces it to continue in its motion 
and makes it describe the arc A ff; from that instant also gravity, 
acting in the contrary direction of the motion, tends to stop it 
The velocity diminishes constantly, and would become nul 
the moment when the ball would arrive at a height equal and 

General Functions of Clocks and Walckes. 


symmetrical to that from whence it started, if there were no other 
forces than those of gravity which would act on the pendulum. 

These forces which exert their action in the contrary direction 
to the motion, are resistances of the suspension and of the air ; it is 
then to these that is due the diminution of the amplitude of the 
pendulum's oscillations. IE these forces could be suppressed, the 
motion would be perpetual. 

54. There are two ~ 
manners of suspending a 
pendulum — by means of a 
knife edge and by means ol 
flexible springs. 

The knife-edge sus- 
pension is made in such a 
manner thai the friction is 
very slight, without its be- 
ing, however, completely 
annulled. This kind of 
suspension can be used in 
regulators whose amplitude 
of oscillation is generally 
small. For this purpose a 
sort of knife blade slightly 
rounded on its edge, and 
working in the interior of 
a hollow cylinder, is fixed 

transversely to the pendulum rod (Fig. ii). The knife edge, as 
the hollow cylinder, should be made of exceedingly -hard 
iterial and thoroughly pohshed. The knife edge can then be 
irded as a pivot of very small dimensions. (We will see later 
on that the work of friction is proportional to the pressure and 
to the greatness of the amplitude. ) 

The spring suspension consists in terminating the upper end 
of the pendulum rod by two short blades of steel securely fastened 
on the other end to any fixed piece. In this system, which is also 
very much used, there exists a loss of force resulting from the 
distorting of the blades. 

55. Concerning Ike resistance of the air, we admit that it is in 
direct relation to the largest transverse section of the body and to 

square of the velocity with which it traverses the atmosphere ; 

Lessons in Horology. 

it depends, moreover, whether its form is more or less tapering. 
The work of this force should be proportional to the cube of the 
size of the amplitudes. 

56. Now that we know the nature of the forces which act on 
the pendulum during its vibrating motion, we can determine their 
work and establish the relation which connects them with each other. 

The motive work IVm developed by gravity during the 
descending half oscillation, is equal to the weight P of the pen- 
dulum multiplied by the projection of the arc B A (Fig. lo) on 
the direction O A ol the force ; therefore, by the length Ab (37). 

We will write then 

U'n. ^ Py.Ab 

The resisting work, that is to say the work of the forces which 
act in the contrary direction to the motion, is composed of two 
distinct forces : 

1st. Of the work of gravity developed while 
the pendulum traversed the half oscillation ascend- 
ing, therefore the weight P multiplied by the pro- 
jection of the arc traversed A B' on the direction 
A; let us represent this work by the formula 

F's '1 W«=Py.Aa 

2d. Of the secondary resisting works arisbg from the resist- 
ances of the suspension and of the air. Knowing the lengths of 
the arcs A B and A B', we find the work of the secondary resisting 
forces by multiplying the weight P of the pendulum by the 
diflerence of the projections Ab — Aa or ab ; we will then have 
Wp = Py. ab 
The motive work should be equal to the sum of the resisting 
works (46) ; we will therefore have 

or substituting 

PXAb = PXAa->rPXab 

57. For the oscillations of the pendulum to preserve the same 
amplitude, it is therefore necessary that at each of these oscillations 
it must receive an impulse whose work should be equal to P X a&, 

58. Since the secondary resisting work increases with the 
amplitude of the oscillations, it is necessary that the impulse, or 
what we should call the work of the maintaining force, should be 
greatest when we wish the pendulum to traverse the largest arcs. 


Genera! Functions of Clocks and Watches. 35 

We see also that the more we diminish the friction of the 
knife edge and the resistance of the air, the less maintaining force 
is necessary. We diminish the resistance of the air by using a 
pendulum ball of high specific gravity, because for such a weight 
the section which traverses the air is smaller. The pendulum can 
also be suspended under a glass from which the air has been 

59. In order to maintain the oscillations of the pendulum in 
clocks we use most frequently motive forces produced by a weight, 
a coiled spring or an electric current. The two first will be the 
subject of a detailed study in the iollowing chapter. 

The Oidllation) of the Balance and <itxai Relation to the Motive Force 

<0. The motion of the pendulum cannot be employed for 
leasuring time, except in instruments which can maintain a fixed 
position. In portable timepieces we utilize the vibratory motion 
of an annular balance, mounted on an axis and furnished with a 
spiral spring. 

61< This spiral spring is a thin blade of metal, of sufficient 
length, wound on itself in the form of the spiral of Archimedes, or 
of a cylindrical, spherical or conical helix. In each case, one of 
the extremities of this blade is fastened to the balance wheel and 
the other to a piece fastened to some part of the watch. 

When we place in a watch movement the balance wheel fitted 
only with its spiral, there is found a position in which the elastic 
force of the spring exercises no influence on the balance. The 
latter is then in the condition of repose. 

When we move the balance from this position in either direc- 
m the elastic force of the spring tends to bring it back to the 
lint of repose ; there are then produced oscillations analagous to 
those of the pendulum. 

This oscillatory motion is very useful for measuring time and 
has the advantage of being suitable for employment in all portable 

62. Suppose A (Fig. 12) the point of repose of a balance 
wheel ; if the latter be moved from that position the angle 
A B = a., and if at the point B it be released, thus allowing the 
elastic force of the spring to act on it, this force will impart to it a 
movement of rotation whose speed will increase up to the point A. 
ling that point, the spiral will exert a force contrary to the 


Lessons in Horology. 

direction of the motion and tending to stop it. It it were possible 
to produce such a movement without there being any passive 
s acting on the balance wheel, the latter would traverse a 

new angle, A O 6 = n., \ 
then would come back to 3, m 
and so on indefinitely. ' 

It is not so in reality, 
for there are a number of 
resisting forces which act on 
the balance and which pre- J 
vent it from arriving at b. I 
These forces are : ' 

1. The friction of the 
balance pivots. 

2. The resistance of the 

3. A loss of force re^l 
J., J, dent in the spiral, the true 

cause of which is not abso- 
lutely defined but the existence of which can be perfectly established. 

These secondary resisting forces have the effect of diminishing 
each oscillation a small quantity, which is represented in the figure 
by the angle B O b. Calling a' the angle ff A, wg have 
B' O b ^ a. — a:. 

If, as we have done for the oscillations of the pendulum, w« 
designate by Wm the motive work exerted by the spiral while th< 
balance wheel traversed the angle a, Wa the resisting work pro- 
ceeding from the spiral during the second part of the oscillatiop ; 
therefore, while the balance wheel traverses the angle a', and Wp 
the secondary resisting work of the passive forces, we would.' 
obtain the equality (46) 

Wm = 

Jtu + Wp, 

The work of the maintaining force should be, both for the; 
balance and for the pendulum, equal to the secondary resisting^' 
work, if you wish to preserve the initial greatness of the amplitude 
of the oscillations ; otherwise expressed, the work of the maintaining 
force should be equal to the work of the force of the spiral while 
the balance wheel traverses the angle a — a.'. 


General Functions of Clocks and Watches. 


63. We can admit that the resisting work increases with the 
amplitude of the oscillations, as we have shown for the pendulum, 
and conclude that more motive work would be necessary to tra- 
verse larger arcs than for smaller ones. 

<4. We use exclusively for motive force in portable timepieces 
the elastic force developed by a spring enclosed in the interior of a 


■ called the barrel. Tiiis piece, generally toothed, turns 
lOund an axis, and this action is conveyed to the balance wheel by 
>ecial mechanism, which we are going to pass rapidly in review. 


lb PuTpoie in the MechBoiiin of Clocki aad Watche*. 

65. The motive force, not acting directly either on the pendu- 
or on the balance wheel, is first transmitted by a system of 
toothed wheels or train of gearings that is called in technical lan- 
guage the wheel-work or the transmission. This force, thus trans- 
irted, is received by a mechanism which is the escaperneni ; it is 

3S Lessons in Horology. 

this last whose function it is to restore to each oscillation of the j 
pendulum or balance whee' the loss of force, 

ff'-. — (K. . 
occasioned by the secondary resisting forces. 

66. When a weight is used as motive force, that weight is sus- 
pended to the extremity of a cord unwinding from a cylinder fixed 
concentrically on the axis of a toothed wheel. This wheel A 
(Fig. 13) gears in a second wheel much smaller than the first and 
which is called a pinion, on which is fixed concentrically a second 
toothed wheel B, which in its turn gears in the pinion &, and so on 
to the last pinion, on whose axis the escape wheel is fastened. The 
same thing takes place when the motive force is that of the spring 
in the barrel. In this case the barrel gears directly into the fit 
pinion a. 

bX, The diSerent wheels of the wheel-work in watches 
the following names : 

1. The barrel. 

2. The center whee! * (large intermediate wheel). 

3. The third wheel (small intermediate wheel). 

4. The fourth wheel (seconds wheel). 

5. The escape wheel (escape wheel). 
The pinions carrying the four last mobiles take the same name: 

as the whee! to which they are riveted. 

68. The mechanical worlc of the motive force is then trans- 
mitted by the wheel-work to the escape wheel. This transmission 
cannot be effected, however, in a complete manner, because part 
of the force is absorbed by the friction of the gearings and of the 
pivots, by the inertia of the moving bodies and sometimes also by 
the defects resident in the gearings. 

69. Beside the transmission of the force, the wheel-work should 
fulfil another function : this is to reckon the number of oscillations 
that the pendulum or the balance wheel executes during a deter- 
mined time and to indicate this number by means of hands on a 
properly -divided dial. We must, therefore, combine the relation 
of the numbers of teeth in the wheels to the numbers of leaves in 

II Id n» 

of the 1 

•WealfB hon lb* KiipnIU 
ointuD (>? Nausbktel. Tbe uunM vt Incxs und of ■: 
ohmleie and ■hmild bt nnlsHid Vy lbs (Iillowtait, ithlcb Bra in bt 
of ihHs two mohtUw 1 " CanWf wheel " for Ihe drat and '■ int«riaf 

General Functions of Clocks and Watches. 39 

the pinions, so as to make this indication conform to the division of 
time. Thus, the center wheel carrying on its axis the minute hand 
should complete one rotation during one hour, and the fourth 
wheel carrying the second hand should make one revolution each 

(The hour hand is carried by a wheel forming part of ail 
accessory wheel-work, which will occupy our attention later.) 


70. Several kinds of escapements have been constructed, 
difiering more or less from each other, but whatever they may be 
their function consists always in restoring to the pendulum or to 
the balance wheel the speed which the passive resistances have 
made them lose. The most perfect escapement will be the one 
which will effect this work by altering as little as possible the dura- 
tion of the oscillation. 

Since the movement of the balance wheel as well as that of the 
pendulum is an oscillating movement, the escape wheel is arrested 

^ during part of the oscillation ; it is only when the balance or the 
pendulum has traversed a determined arc that the wheel becomes 
free and is put in motion. During this time it acts either directly 
on the balance, as in the ' ' cylinder ' ' escapements or the ' ' detent, ' ' 
or on an intermediate piece, as in the "anchor" escapements. 
After having traversed the angle of impulse determined, the wheel 
arrested anew until another disengagement. The manner in 
ich this arresting is produced differs according to the kind of 
Tl. In most of the escapements the action of each tooth of the 
wheel corresponds to two oscillations of the balance wheel or pen- 
dulum. Thus, in a watch, the balance wheel executes 30 oscilla- 
during one complete revolution of a wheel of 15 teeth ; in a 
, the pendulum makes 60 oscillations during one revolution of 
escape wheel of 30 teeth. 

72. To recapitulate, the study of the functions of horological 
ichanism can be divided into four principal parts, which are ; 

Power — study of motive powers. 
, Transmission — study of wheel-works and gearings. 
. Reception — study of the escapements. 
, Regulation —study of regulating and adjusting. 

^Hfa ai 

Maintaining or Motive Forces. 

The VlafiA u a Motive Forcr. 

T3. We will adopt in the beginning as units in the calculations, 
the millimeter as unit of length, the gramme as unit of weight and 
of force, which gives us for the unit of mechanical work the 
grammillimeter. We will choose the second as the unit of time. 

74. Among all the forces which are used in horology in order 
to maintain the oscillations of the pendulum, the weight is at once 
the most regular, the most simple to obtain and the one whose 
intensity can be regulated with the greatest facility. 

75. If a certain weight P (Fig, 13) is suspended at the end ot^ 
a cord wrapped around a cylinder the radius of which increased by , 
half the thickness of the cord is equal to r, the work of this force while; 
the cylinder executes one revolution will be expressed by (38) 

Z' X 2 T r. 
Dividing this work by the number N of oscillations that the, 
pendulum executes during one revolution of the cylinder, we will 
have as quotient the mechanical work developed by the weight 
during one oscillation of 'the pendulum, thus : 

We know that a part of this mechanical work is lost during its J 
transmission to the pendulum : calling W^ this last work, we.J 
should have the equality 

W /• — Ifl ^ Wp, 
in which we will replace W Phy its value, thus : 

We see then that the determination of the work which the 
pendulum receives at each oscillation ( I'l^ ) depends also on the 
knowledge of the work lost during its transmission by the wheel- 
work and the escapement. We understand, consequently, the | 
diiliculty that there is to determine the motive work, since this j 
work does not depend alone on the weight and on the dimensions j 
of the pendulum but also on the resistances to be overcome during 
an oscillation. 

Here are. however, two calculations taken from practice which \ 
will aid in more firmly fixing the ideas on this subject ; 


Maintaining or Motive Forces. 4! 

M. First Calculation. — The motive weight of a regulator 
beating seconds is 2000 grammes ; this weight is suspended at the 
end of a cord which unwinds from a cylinder, with a radius of 
15 millimeters. What will be the work produced by this weight 
during the unit of time ? 

The mechanical work efiected by the weight while the cylinder 
b executes one revolution will be 
■ EocKj X 2 IT X 15 -= "88496 gr.m. 

1 A wheel.<4 is fastened to the cylinder (Fig. 13) gearing in a 

pinion which carries on its axis a second wheel B, which in turn 
gears into a pinion &, this last pinion carrying on its axis the minute 
hand should then execute one revolution an hour. The numbers 
of teeth and leaves of these moving bodies are distributed in such 
a manner that the pinion f> executes 45 turns while the cylinder 
makes one ; consequently, one revolution of the cylinder takes 
45 hours or in 

45 X 60 X 60 = 162000 seconds = A^ 

We will then obtain the work produced by the weight during one 
oscillation of the pendulum, by the application of the formula, 


» /> - -V^ - f^Z -= ^-'^3 Er.T 

We will show the manner of calculating the work lost during 
Bie transmission when we treat of the questions of frictions, of the 
inertia of the wheels, etc. ; for the present, let us admit these calcu- 
lations as made and adopt for this special case the value 

ItVu = 0.413 gr.m, 
e will then have 
tfm — ICb = Wp , 
1.163 — 0,413 = 0.7s gr.m. 
The weight of 0.75 grammes, exerting its action on a dis- 
ice of one millimeter, is then sufficient to keep up the oscillations 
01 a pendulum whose weight is about 6500 grammes. The ampli- 
tude of the oscillations is 2° 6'. 

7r. Although that which follows is a little outside of the problem which 
we have just solved, let us profit, however, by the data that we possess to 
calculate further the angle 

B O A — B' O A (Fig. 10). 
'his adjunct to the preceding solution does not, moreover, lack in interest. 


Following an equalion previously established (56), the work of the force | 
capable of maintaining the oscillations of a pendulum was expressed by 

IVp = PX a 6. 

" ^ = -6500" = 0.0001154. 
The length of a simple jjendulum beating seconds is about 994 milli- 
meters for our latitude.* Let us suppose that the entire weight of our pen- 
dulum is assembled at a single point, the distance from the center of gravity 
to the center of suspension will then be equal to the length of a simple pen- 
dulum beating seconds. We will have 

^i = 994 — 994cos/^05 


From Fig. 10 the difference A b ~ 
tracting then the two foregoing equations, 

a 4 = 994 cos .rJ £' - 
whence follows 

since the angle A O B\i equal, in this case, to half of 2° (/, which is 1° 3', 
we can write, after having completed the calculation of the second member 
t& the equation : 

Cos A OB' — cos 1° 3' = 0.000000116. 

In order to determine the value of the angle A O B ^ A B', we can find 
in a table of natural trigonometrical lines the difference between the cosineSn 
of the angles 1° a' and 1° 3'. This difference is 0.0000053 ! ws will theO' 
have the proportion, 

then AOB — AOB'= i.-f. 

78. Second Calculatimt. — A clock from the Black Forest, such 
as those that were manufactured in iai^e quantities during the years 
between 1S40 and 1850, runs under the action of a weight 
625 grammes. This weight descends in 24 hours from a height of 
1350 millimeters. What is the work produced by this force during 
one second ? 

The work produced during the descent of the weight will be 
W= 635 X IJ50 
in 34 hours ; during one second it will be 24 X 60 X 60 = 
times less ; therefore 

* LiUiuilc ct Neuoliilil 

Maintaining or Motive Forces. 43 

We see that this clock requires a much greater mechanical work 
than that of the regulator of the preceding example. This difference 
becomes still more obvious if we compare the two pendulums. 
The weight of the pendulum of the last clock is only 8 grammes, 
while the pendulum of the regulator weighs 6500 grammes. 

Although we could not, at this time, compare two clocks, 
whose pendulums have neither the same length, nor the same 
weight, nor the same amplitude of oscillation, we note, however, 
that the regulator requires much less motive force than a small 
^ clock of the Black Forest. 

The Barrel Spring u n Motive Force. 

79. These springs are thin blades of properly- tempered steel ; 


^H^they are of a sufficient length and coiled up in spiral form i 

^■'interior of the barrel. One of their extremities is fastened to the 

^H- wall of the drxim and the other to the hib, which is a cylindrical 

^^Vpiece adjusted on the arbor of the barrel or forming part of it, 

^^"When one holds firmly either the barrel arbor or the barrel, and 

^^ causes tlie one of these two pieces left free to turn, the spring 

begins to wind around the hub and manifests a certain force from 

its extremities, which tends to bring it back to its first form. When 

the arbor is made fast, the force displayed by the spring has then 

the eSect of causing the barrel to revolve. 

80. The place occupied by the spring in the interior of the 
barrel should be equal to half the disposable space. 

81. Measurement of the Force of a Sprlnf. The force developed 
by the spring is susceptible of measurement. For this purpose let 
us adjust on the barrel arbor a graduated lever arm, along which a 
certain determined weight can slide. While holding the barrel in 
the hand, let us set up the spring to the point that we wish to 
study, one turn for example ; let us endeavor then to produce 
equilibrium by sliding the weight along the lever arm. When the 
two actions, that of the weight on one side and of the spring on the 
Other, neutralize each other, equilibrium is produced, and it is then 
evident that the efTort displayed by the spring is equal to the 
eflect produced by the weight. This last effect will be perfectly 
determined when we know the size of the weight and the length 
of the lever arm, at the extremity of which it exerts its action. 
We know that in mechanics the moment of a force (4a) is the 

—product of the intensity of this force by its lever arm. 


Lessons in Horology. 

The moment of the iorce of the weight will give us then the 
moment of the force of the spring. 

82. If the lever of the preceding experiment has not its center 
of gravity on the axis, it will still be necessary to take account of 
the effect produced by the weight of this lever, which cannot, 
practically, be reduced to a simple geometric line. In order to 
determine this we must find the distance of the center of gravity of 
the lever from the axis, and multiply this value by the weight of 
the lever. We then add this product to the moment of the force 
previously obtained. 

Let us suppose, for example, that a weight of 20 grammes, 
suspended at the extremity of a lever arm 200 millimeters long, makes 
equilibrium with the elastic Iorce of a barrel spring. The product 

20 X 200 = 4000 
represents the moment of the force exerted by the weight. 

If, moreover, the weight of the lever is 7 grammes, and the 
distance from its center of gravity to the center of the arbor 143 
millimeters, the moment of the force e.xerted by the lever will be 

7 X 143 = io°i- 
Adding this value to the moment of the force of the weight, we 
obtain the moment of the force of the spring that we will designate 
by F, then 


- 1000 = 500Q grammes 

the approximate value of the moment 

in round numbers. 

of the force of the spring in a watch of 43 millimeters (ig hnes). 

Let us remark that generally these levers are furnished with 
counter weights combined in such a manner that the center of 
gravity is found on the axis. 

83. The number 5000 that we have just obtained, signifies 
that the spring considered is capable of making equilibrium with a 
weight of 5000 grammes suspended at the extremity of a lever arm 
equal to the unit of length, therefore i millimeter (Fig. 14). 

84. Examining in this manner the force of a spring, we will 
prove that it varies very much according to the number of turns 
that it is set up. Experience proves in fact that the moment of the 
force of a spring being, for example, at its maximum point of 
tension, 5000 grammes, this moment constantly diminishes, and 
will not be more than about 3400 grammes when the barrel will 
have executed four rotations around its axis. 

Mainiaining or Motive Forces. 45 

85. We understand then thai the imperfections of the primitive 
watches being known, the ancient horologists should have sought 
means for correcting the inequality of the action of the motive 
spring, and that for this purpose they should have invented the 
ingenious arrangement of the fusee, which will be explained 
later on. 

This corrective is really almost entirely abandoned, and is 
seldom used except in marine chronometers ; in pocket watches it 

has become useless in ]>rDportion as the improvements in the con- 
struction of escapements have come into use, and as the isochronism 
of the oscillations of the balance wheel has been obtained. 

ITheoretica] Study of the Moment of a Sprin£'j Force. 
86. Coefficient of Elasticity. When a body receives an exterior 
effort, the molecules which compose it tend to follow the direction 
of this force ; they approach each other or separate themselves, the 
one from the other. The result is a force equal and opposite, 
iFhich tends to make the displaced molecules recover their former 

This property, common to all bodies in different degrees, is 
illed their elasticity. 


LessoTis in Horology. 

According to the effort exerted, the molecules approach or 
leave each other ; the first case is an effect of compression or 
contraction, the second is an effect of tension. 

ST. The reaction is always equal to the action ; we can then 
measure the elastic force of bodies by the exterior effort which 
is applied to them. The following experiment wilt explain this 
assertion : 

88. Let us secure one of the extremities of any vertical rod, 

to the other extremity we suspend a weight (Fig, 15). This 

rod from that moment undergoes a certain elonga- 

BHSMl'MI l^'on, and we can prove that the molecular efEort devel- 

1^^^^ oped is equal to the weight producing the elongation. 

The elongation of this rod will depend on the size of 

the force P, on the length of the piece in its natural 

state, on the cross section of this piece being assumed 

the same throughout, and finally on the material of 

which it is composed. 

By experimenting on a rubber band we can see 
that under the action of a force P, the transverse section 
of the body diminishes at the same time that the elonga- 
tion is produced. This regular diminution on almost 
the entire length of the band does not take 'place 
uniformly near the two points of fastening. Therefore, 
it is necessary to take the elements for the experiment 
sufficiently removed from these points in order to 
eliminate a source of error which would influence the 
final result. 

Let us take again, for example, a rod of iron, 
whose transverse section is i square mm. ; we have 
measured the distance between two marked points 
sufficiently removed from the points of fastening ; let L 
be this length. We suspend from the lower end of 
the rod a weight P, and we measure anew the length 
between the two marks ; we obtain then an elonga- 
fig. IS tion /. Experiments made in this manner have de- 
monstrated that, provided the load P does not surpass 
a certain limit, / remains proportional to the load. 

Supposing now that the experiment were physically possible, 
let us determine what should be the load P that could produce 
an elongation equal to the original length L. We call thia 

Maintaining or Motive Forces. 

particular value of P the coefficient of elasticUy of the body ; we 
will designate it by the letter E. 

The elongation being proportional to the load, we have 

Thus, in the case of an iron rod, whose original length L 
was looo millimeters, if we suspended from it a load P of looo 

: will find an elongation of 0.05 mm., which gives 

as the coefficient of the elasticity of iron, 

89. The elongation / is inversely proportional to the transverse 
section of the body ; thus for a section of surface s the formula 

I above will become „ . 

90. When the coefficient of elasticity is known, it is easy to 
determine the value of the force exerted by the molecules of a 
body subjected to the action ol an exterior force by the relation 

p — --'-' 

'he fraction j represents the elongation per unit of length ; this 
.ction should remain very small for this formula to be exact. 
The quantities £, s, L are constant ; P and / vary together. 

The same formula expresses the relation which connects a 
force P of compression to the contraction /. which results from the 
action of this force, when the piece compressed does not bend. 
We will give then to P and to I the signs + and — , -|- for the 
forces of tension and the elongations, — for the forces of com- 
pression and the contractions ; the formula then becomes general. 

91. Variation of the Coefficient ol Elasticity. All watchmakers 
know that after having forged a piece of brass, the elastic force of 
the metal is increased. In hammering this body one diminishes its 
volume, but one cannot change its weight ; the molecules which com- 
pose the piece are forced together, and the specific weight of the 
metal will be increased. This simple fact shows us that the coefficient 
of elasticity of solid bodies should vary with their specific weight. 

When a watch (not compensated), regulated to a certain 
temperature, is exposed to a higher temperature, it loses about 

48 Lessons in Horology. 

10 seconds in 24 hours for each degree centigrade. The spring is 
expanded by the effect of the increase of temperature, its mole-- 
cules are separated from each other ; the specific weight of the 
metal has diminished at the same time as its coefficient of elasticity.. 
The reverse takes place when the watch is observed at a lower 
temperature than that to which it had been regulated. 

It does not appear that the coefficient of elasticity of sted 
undergoes a great variation by the effect of tempering and that of 
reheating. A piece of steel in fact changes its dimensions ^ 
little by tempering. It has been proven that by tempering va- 
water a piece of steel stretches about , j^j of its original length, 
but that this elongation is lost when the piece is reheated to the 
blue color, the specific weight of the steel not being modified, the 
coefficient of elasticity retains the same value as that which it. 
possessed before tempering. 

92. We give here a table of the coefficients of elasticity of 
some bodies employed in horology. The figures below are taken 
from the " Almanac of the Bureau of Longitudes."* 

VkIuw of the Coeffidenti of Elutidty, E. 


Bronze : 90 Coppei 



Phosphorous ozjo 

Laveissiere 9061 

Copper 12jM9 

Berry Iron 2097a 

Rraw ■ / 31 Zinc 9277 

«"^' 168 Copper 9395 

f 18 Zinc 1 

German Silver : -j 61 Copj 


I Nicf 

Gold 8132 

Palladium '1759 

Platinum 17044 

,rid»dP,.,i„™: {»■/«"»„} 



ed In kiloKrinimeB 1 


■Itclty In 1 
menfly eap 

fore laSKlOO 

Maintaining or Motive Forees. 


93. Limit of Elasticity. If we submit any rod to the action of 
difierent loads, we note that as long as the load does not exceed a 
certain limit, proportionate to the transverse secdon of the body, 
the rod resumes its original length, after the removal of the weight. 
By increasing the weight so as to pass this limit, the elongation only 
partially disappears or perhaps does not disappear at all. This 
limit is called the "limit of perfect elasticity" of the body con- 

sidered. If we continue to increase the weight, the elongation 
becomes more and more apparent and at length a "rupture" is 

The limit of perfect elasticity is very slight for certain metals, 
such as lead, red copper, aluminum, etc. Iron, even, does not 
possess a very great limit ; on the other hand, steel, when it is 
tempered, increases its limit of elasticity by suitable reheating. 
This reheating, known under the name of "spring temper," cor- 
responds to the bright blue color. 

It would he of great use in horology to know the exact value 
of this limit of perfect elasticity of hardened and tempered steel ; 
these experiments have not yet been thoroughly studied and the 
data is consequently lacking. For the present we will confine our- 
selves to the results with which the practice of horology furnishes us. 
94. Homent of tbe Elastic Force of a Spring: Subjected to a 

FiexiOQ. Let A^ B^ be a spring of circular form, of rectangular 

Lessons in Horology. 



section, of thickness e and height k. Let us imagine 
blade of spring be divided in the direction of its length into a 
certain number of fibres, one of which, especially, situated in the 
middle of the body, is called the "neutral fibre" for the reason 
that it does not change its length when the spring undergoes a 
flexion. When this blade is bent in such a way that the radius of 
the neutral fibre diminishes (Fig. 16), the fibres interior to this 
undergo a shortening, while the exterior fibres are lengthened. 

Let it V be the distance of any fibre from the neutral fibre. 
+ 11 if the fibre is on the exterior and — vM it is on the interior rf 
the neutral fibre. If r„ represents the radius of the neutral fibre 
the unchanged position and e the angle that the two radii ending' 
at the extremities A^ and B^, form between them, we will have the 
length /-o of the neutral fibre by the equation 

io = r^e 
and the length Z,', of any fibre whatever whose radius is 7 
will be 

L', = (r„ + z') 6. 

If now, one of the extremities of the blade is fastened and we 
bend the other, making it traverse an angle ± «, the radius of the 
neutral fibre will diminish if a. is positive, that is to say, if it adds to 
the angle e. If, on the contrary, the extremity Bq is bent in the. 
opposite direction, the angle fl becomes smaller and we have ii 
this case a negative : the radius of the neutral fibre will increase. 

The length L^ of this fibre has not changed by the flexion 
we will have then in this new position, r being the radius 
changed position of the fibre, 

Lo =r (B + a), 
we then have 

r. e = r (B + d), 
from whence 

The fibre taken whose length is L\ has become 

L', = (,r+!>) (fl + . 
Replacing r by the value above, we will have 

^'. = (|R:\ +•)<' + "• 

and in working out 

Maintaining or Motive Forces. 51 

the elongation / of the fibre considered will be obtained by taking 
the difference between lengths L' and L\ then 

i' — i'o = / = r„ e 4- 1- e + ;- a — fo s — K e 
and simplifying (,) i = v 0.. 

This elongation is positive ; but it will become negative for v 
negative ; that is to say, for the interior fibres there will be a 
shortening. There will also be a shortening if v is positive and a 
negative. If these two values are negative, their product is posi- 
tive and we have an elongation. Let us remark that the elonga- 
tion / is independent of the radius r^ of the neutral fibre and that 
consequently the spring can be of any form. 

95. Let us now determine the moment of the force exerted by 
two opposite fibres, situated at equal distances + v and — v from 
the neutral fibre and let us suppose that the flexion of the blade 
may have been effected preserving the center O to the changed 
position of the spring ; that is to say, that the blade may have 
taken the position A B (Fig. 16). 

The exterior fibre, which has been lengthened by the flexion, 
will tend to return to its first length and will act with a force P 
whose value is represented by (90) 

which we can replace / by its equivalent v a. (94), which gives us 
le interior fibre tends to lengthen out and will exert this same 
ce in the opposite direction, therefore 
— ^- 

The moment M P o\ the simultaneous effort produced by the 
the action of these two fibres will be equal to the sum of the pro- 
ducts of each of the two forces by their respective lever arms r \ v 
■ V. Therefore 

M P ^ P {r-^v) — P {r~v). 

(3) MP^2 Pv. 
This value is then' independent of the radius of curvature of 
'tL spring, that is to say of the distance from the exterior attaching 
int of the blade to the center of the barrel. 

Lessons in Horology. 
Replacing in the equation (3), P, by its value (a) v 

Let us now regard the section s, of the fibres considered. 
The cross section ol the spring being imagined rectangular, of 
height A and thickness e we will have, supposing first that the blade i 

is divided into a definite number of fibres, 10 for example, the 

section of one of these fibres 

J = 0.1 f X A, 

since the thickness of one fibre will be in this case the tenth part of 

the total thickness of the spring. 

Let us admit, what is not absolutely exact, that each separate 

fibre acts through its middle part, that is to say, that the distance V 

from the middle of the fibre nearest to the neutral fibre be 0.05 C' 

(Fig. 17), the distance of the middle of the second o. 15 e, that of 

the third 0.25 e, for the fourth 0.35 e and then for the fifth o.i 

Since in the equation (4) the term v is to the second power we 

should raise each of the five preceding values to the square and i 

find the sum of them. We will then have 

= 0.0025 '* 
= 0.0225 '' 
= 0.0625 *' 
= 0.1225 ^* 
^ o, 2025 e* 

= 0.4125 e* 

1st fibre w =: 0.05 e 

. v-^ = 0.05= 

2d " 1/ =^ O.ISf , . 

. v^ = 0.15^ 

3d " V = 0.25 e . . 

■ w" = 0.25* 

4th " V = 0.35 e . . 

■ ■o'^ = 0-35' 

5th '■ V = 0.45 e . . 

. v^ = 0.45 » 

The sum of 


MahUaining or Motive Forces. 53 

Replacing now the values determined of s and of i'* in the 
equation (4) we will have 

t- ,, „ £ a 0.1 « A. 0.4125 f' 
Sura M P ^ 1 -. — - — 

This formula represents the moment of the force of all the ten 
fibres considered, therefore of the endre spring, while the formula (4) 
gave the value of the moment of two fibres only, the one interior 
and the other exterior, to the neutral fibre. Designating by F the 
preceding sum, we will have, by performing the operations 
indicated „ c . 1 

(5) F= °-">^'-'- . 

We have obtained the coefficient 0.0825 by dividing the blade of 
the spring into 10 fibres ; ii we had supposed it divided into a 
very large number of fibres, we would have arrived at a value 
very nearly approaching 0.08333, say, ^. We would have then 
this case. 

(6) '' = — ;fi-'^ 

96. We have arrived at this last form, which is the exact 
'one, only by approximation. 

Integral calculus furnishes us a means of effecting the above 
calculation with an absolute exactitude and ior an infinite number 
of fibres. (*) 

take up again the equation (4) 

MP = 

ignating the inliDitely small thickness of a fibre by dv 
the section s 

s ^. dt/.k; 

e~J "■'" = ! — 

Utlne the 
irtntlo tl 

Replacing f by J r, that 

Lessons in Horology. 

isy, taking the integral between the limits^ 

-^ = A 2 "■ 

97. In the preceding formula, a is the angle which we have 
made the free extremity of the spring describe from the position where 
the elastic effort is null, to the point which we wished to study. 
Thus, when we have turned the tree extremity of the spring one 
revoiudon, the original number of revolutions will be increased by 
r, etc. We can then estimate the angle a by counting the number 
of turns which the spring is wound up at the moment considered, 
not forgetting to deduct from tfiis figure the number of turns which 
the spring makes if placed unconfined on a table. Let n be tbiS'— 
diSerence, we will have H 

we can write the formula (6J H 

98. On calculating the moment of the force of a barrel spring \ 
by means of this equation, and on then comparing the result 
obtained with that which the experiment gives (S3), we generally 
find a slight difference. This difference proceeds essentially from 
the following causes : 

ist. As we have already stated, the value of the coefficient 
of elasticity of the spring with which we are engaged could be 
perceptibly different from that which we have admitted in the 

ad. It is difficult to measure exactly the thickness of the spring s 
a slight error will give a considerable difierence in the result Thus* 
for a spring of o. i3 mm. an error of -j^ °^ ^ millimeter wl 
influence the result one-sixth. 

3d. The transverse section of the blade is rarely a perfeel 
rectangle ; the spring is often concave on the outside and convex aU 
the inside. 

4th. The calculation supposes the spring to be perfecdy freCj 
but complicated effects are produced when it is shut up in the barrel. 

When it is wound around the hub of the arbor there is only a I 
certain length of the blade which is freed from the coils pressed 
against the drum of the barrel. The moment of the force should 
therefore be calculated according to the length of the blade freed. 


Maintainmg or Motive Forces. 55 

5th. When the spring is wound up to a certain point, the coils 
of which it is composed deviate from the circular form and spread 
out to one side ; there is thus produced a decomposition of force, 
one of the components of which is directed towards the center of 
the barrel and is transformed into friction. We can add a similar 
defect which is produced at the center and which on combining 
with the exterior fault can diminish, or, in certain cases, increase 
the moment of the force of the spring. 

6th. Considerable friction is produced between the coils of 
the spring ; the oil which we are obliged to use to reduce friction 
produces a slight effect by its adhesive force. 

99. Example for the Numerical Calculation of the Formula (7). 

The dimensions of the spring for a watch 43 mm. diameter (19 lines) 
being the following : 

Thickness, ^, =0.18 mm. 
Height, hy =- 3.6 mm. 
Length, Z, = 650 mm. 

to calculate the moment of the force of this spring. 

When the elastic effort of this spring is nothing, that is to say, 

when it is placed perfectly free on a table, it makes 5 turns. Coiled 

in the interior of the barrel and pressing against the drum, it makes 

14. The development of this spring being 6 turns in the barrel, a 

half turn is given for safety, and we will have, according to what 

has been said, 

« = 14 + 5.5 — 5 = 14.5 turns, 

when the watch is completely wound up. 

Let us take the coefficient of elasticity, E = 23000000. The 
formula (7) can be written thus : 

-, E h e^ le n 

replacing the letters by their values, we have 

„_ 23000000 X 3-6 X o.i8^ X 3-1416 X i4'5 

6 X 650 

Effecting the above calculations we find that 

F = 5640 gr. 
for « = 14.5 turns. 

Lessom in Horology. 

The simplest way of effecting the above calculation is by means 
of logarithims. We give below the method of such an operation : 

Log. E ^= Ic^. 230000CX) = 7.3617278 

log. e» = log. 0.18* 
log. h = log. 3.6 
log. , ^ log. 3.1416 = 
Log. numerator 
Log. L — log. 650 = 2 
+ log. 6 = 

- 0.7658175 - 3 

= 0.5563025 

^ 0.4971499 

6. 1809977- J 
8129134 m 
7781513 ■ 

Log. denominator 3 

Log. numerator 6 

— log. denominator 3 

5910647. ■ 



+ log. K = log. 145 = I 

5899330 = log. 38S.985 
161 5680 


7513010 = log. 5640. 

The preceding calculation shows that the moment of the force 
ol the spring is 388.985 gr. for a number w ^ i. For « ^= 14.5 
ii is 5640 gr. When, on account of the running of the watch, the 
barrel has made one turn, /( will have diminished one unit and will 
only have 13.5 as value ; the moment of the force of the spring 
will have diminished 388.985 gr., or, in round numbers, 389 gr. 
We can then form the following table ; 

/" f or « ^ 14.5 = 5640, the spring is entirely wound up, 
/^, '■n^i3.5^525i| the barrel has made one turn. 
J-'j " K r= 12.5 = 4862, the barrel has made two turns. 
F'f " n =^ II. 5 ^ 4473, the barrel has made three turns. 
F't " n = 10.5 ;^ 4084, the barrel has made four turns. 

100. Inequality of the Elastic Force of the Spring:. The 
moment F of the force of a spring is then greater when the watch 
U completely wound than when it is about to stop. This fact has 
been already demonstrated to us by experiments (84). 

It b necessary to confine this inequality of the motive force 

witilin the narrowest limits possible. Let us note for this purpose 

tin the numerical calcularion of the formula {7) we have succes- 

^ replaced n by h — r, « — 2, w — 3, and a — 4 ; in this last 

I IflW Ibe traldi is at the instant when it is about to stop, if the barrel 

I with stop works. But the ratio between n and n — 4 

1 is smaller, it will be proper, in order to diminish 

t {nqnafityof the force, to give to the number of turns, a, of 


Maintaining or Motive Forces. 57 

the spring, as great a value as possible. The following demonstra- 
tion will better explain this idea. 

lOL Since we can use springs producing the same initial moment 
offeree ^0' ^"^ whose dimensions and number of turns, «, are 
different, we understand that F^ may vary in certain cases much 
less than in others. 

Let us suppose, for example, two springs producing, when 
wound up, the same moment of force F^ = 4000 ; the first having a 
number of turns n = 10, the second a number » = 20. For the 
firat we would have in the formula (7) the value 

and for the second this same valut 


When the two barrels will have executed one revolution, the num- 
ber of turns of the first will be n, = 9 and that of second »i = 19. 
will then have successively : 

First Case. 
F^ ^ doo X 10 ^ 4000 
F^ ~ 400 X 9 — 3600 
F^ = 400 X 8 = 3200 
Ff = 400 X 7 ^ aSoo 
Fi ^= 400 X 6 = 2400 

Second Case. 
F^ = 200 X 20 ^= 4000 
Fi = 200 X 19 = 3800 
Fj = 200 X 1 8 — 3600 
F^ = 200 X 17 = 3400 
/^, — 20a X 16 = 3200 

It is thus easy to see that the moment of the force diminishes 
in the first case much more rapidly than in the second, and that, the 
force of the second spring approaches much more a constant value 
than that of the first. 

It is best, then, to give to the number of turns, a, the greatest 
value possible. 

For a given spring this number cannot, however, exceed a cer- 
tain value determined by the limit of perfect elasticity of the steel, 
which cannot be exceeded without setting or breaking the spring. 

This limit depends on the elongation per unit of length ~ of the 
exterior fibre. We have had (94) 

h can be written 


Lessons in Horology. 

In the preceding numerical example we have had the following 
values : 

v^\e^ 0.09, a = 3 « fi = 14.5 X 2 ■». Z = 650 mm. 

We will then have 

J_ ^ 0.09 X 14-5 X 2 ■• 

L 650 

and on performing the calculations 

X = °''"^"' """■ 

We can admit this value of -j- as the limit allowing sufficient 
security, and established by practice. 

102. It must not be forgotten, however, that the nature of the I 
steel, the manner in which the springs are manufactured, hardened | 
and tempered, can materially modify this limit. 

The springs are hardened and tempered in circular form, witll | 
about 100 mm. radius ; they are then worked in a spring tool and 
by this operation the fibres undergo quite an unequal elongation, 
since, from the first circular form, they pass to a spiral form whose 
radii of curvature for the interior coils are much smaller than thosM , 
for the exterior coiis. 

It follows, from this operation, that the exterior fibres : 
elongated more in the inner coils than in the outer ones. It is4 
for this reason that the springs break more often interiorly than 1 

103. The form which a spring has on leaving the hands of the 
maker is very variable and it can be with difficulty represented 1^ 
a general equation. 

' This primitive form is not preserved after the watch has run a 
certain time : the spring "gives " a httle at first and finally ends by , 
taking a form which it keeps permanently. This last form is the one | 
ft-hich should be taken as the starting point from which to determine 
the angle inn giving the degree of flexion in the formula (7). 
Starting from this position, we can admit that the elongation per I 
unit of weight -3- is much the same for the whole length of the spring, j 

104. Length of the Spring. Since we cannot unfold a spring, , 
in order to measure its length, without modifying its interior strue- j 
ture, it is convenient to have at our disposal a simple formul 
enabling us to calculate the value L. 

Supposing the spring coiled in the interior of the barrel ; 
will admit that the radius extending to the interior coils may t 

Maintaining or Motive Farces. 


equal to ^ ^ in the position of the spring at rest, R being the 
interior radius oi the barrel. We can, without great error, substi- 
tute circumferences for coils and obtain the length of the spring by 
multiplying the mean radius, 

by 2 w N', on deciding to designate by N' the number of coils in the 
spring when it is pressed against the side of the barrel. To this value 
must be added the length of the end of the spring which is detached 
■om the coils so as to be hooked to the hub, which is about 

; would have, therefore, the length 

L ^ -i -^ R (I N' + \). 

We have, for example, in the calculation of the length of a 
iring for a watch of 43 mm. the following values : 
R = 8.8 mm. and N' = 13 coils, 
[tepiacing the values, we will have 

Z = 2 X 3-1416 X 8.8 (J X 13 + i); 
I whence 

Development of a Spring. 

I05. When a watch spring is put in the barrel it is wound on 
itself and forms a certain number of coils, the outside one of which 
presses against the side of the barrel and the succeeding ones against 
each other, taking the form of a spiral of Archimedes, with grada- 
tion equal to the thickness of the spring. The inner end of the 
blade is disengaged abruptly from the coils and is fastened to the 
hub of the arbor. 

In order to "wind" the spring, we can hold the barrel and 
turn the arbor several turns until the spring may be entirely wound 
around the hub, with the exception of its outer end, which remains 
fastened to the side of the barrel. 

It is evident that the number A'' of revolutions which the arbor 
has been able to make is equal to the difference between the num- 
ber of coils that the spring has in these two extreme positions. 


Lessons in Horology. 

Let N', be the number of coils which the spring makes wt 
is pressed against the side of the barrel ; N" the number of coils 
when wound around the hub. We will have then N^= JV" — A". 

In order to simplify the calculations, let us neglect the inner 
and outer ends of the blade which are disengaged from the coils, and 

admit that the space occupied by the spring in the two positions 
a cylindrical volume. Let us designate, moreover by 

/?, the interior radius of the barrel ; 

r, the radius of the hub ; 

r", the radius extending to the interior coil of the spring when 
it is pressed against the side of the barrel (Fig. i8) 

r", the radius extending to the outer coil of the spring when 
is wound around the hub (Fig. 19) ; 

e, the thickness of the spring. 
We can write 

' = and 

i consequently 

(i) A'= N" — N' = 

We can find the value of r" 

^H and CO 

^^B We can find the value of r" in functions of R, r* and r 

^^v observing that the surfaces occupied by the spring in the two poai- 

^H tions are equivalent. 

^H When it is pressed against the side of the barrel, this surface b 

^L 5 = » (^'--^^'), 

Maintaining or Motive Forces. 

when it is wound around the hub 

5 = ir (.*» -»•')■ 

r*' — r' = J? ' — r", 
1 whence 

^ V R^ 

hibsdtuting this value of r" in the equation (i), we have 
(J) N = ~ (^ R^ ~ ^^ ^ r^ - r - R + r'^. 

106. Let us note that when \ 
tquation „ 

: obtain the value of € in the 

we find that it differs from the real value, which is always less. This 
difference arises from the fact that on account of certain inequahties 
of the spring, the coils of the blade do not strictly superpose, 

107. Maxiraum of iV in Terms of r'. The equation (2) 
indicates that for a barrel whose interior radius R, the radius of the 
hub r and the thickness of the spring e, are determined, the num- 
ber of turns of development N varies with the radius f' extending 
to the first inner coil of the spring in its state of rest. This last 
radius, r', depends on the length of the blade. 

Let us apply, in the first place, this formula to a numerical 
mple and use the following data ; 

Interior radius of the fjarrel . . ^ ^ 3 

Radius of the hub r ^ i 

Fraction — ^=13, 

The equation (2) will become, after replacing the known 
uitities by their values, 
N^ I 


- r"" 

The smallest value that the radius ■/ can have is 
r' = r = \ 
lod its greatest value may be 

r' ^R = i. 

In the first case the spring has a number of coils sufficient to com- 
etely fill the space between the side of the barrel and the hub, and 
n the second this number of coils is nothing. The reasoning shoiv^ 


Lessons in Horology. 

that in these two extreme cases the development of the spring' woa 
be nothing, which the apphcation of the formula (2) also proves. 
Replacing successively in this same formula r' by 1,1, i.; 
1,3, etc., up to 2.9, we can form the following table : 





































a. a 










One sees that the maximum number of coils in the develop- 
ment ol the spring is given by a radius r' equal to about 2.2. 

In practice, | of ^ has been adopted as the value of r' for the 
reason that the regularity of the power from beginning to end 
increases with the length of the spring. 

108. The chIcuIus enables one to determine the exact value of the 
maximum of N in function of r'. 

Let us take up again the equation (i) : 

N=\ (^' R^ - r" -i- r' - r - /i + r'), 

N and r'. Let us differentiate this equation 
J?' - r" + r' = ^, 

in which the two variables 
by placing 

d N= ^ {ki* dz + d r'). 

Replacing .; and d a hy their values, it becomes 

f JV= i ( 
• \ 1 


Maintaining or Motive Forces. 
qnating thb derivative to zero : 

: VR* - 

Ijtaising to the square we have 

(3) .'=V^4^. 

Substituting this value of r' in the equation 

We see that the tnaximum oS N o 

s when one has 

— v^ 

Since it is the custom, in practice, to make the radius of the hub equal 
to one-third of the interior radius of the barrel, we can place 
Jf = 3 and r = i 

and we will have 


P 109. In order to represent graphically, the equation (2), let us 

rder it to two rectangular coordinate axes (Fig. 20) and lay oS on 
&e axis of the abscissa the values of r ', and on the axis of the 
ordinate the corresponding values ol JV, and connect the points 
thus obtained by a curved line. 

One sees that on making the unoccupied space of the barrel 
equal to the part occupied by the spring one does not obtain the 
Vmaximum turns of development of the spring. 

110, If one divides the interior of the barrel, giving 
^ ^ to the part occupied by the spring, 
^ R for the empty space, 
i R for the radius of the hub. 

N= \ C/F^T^ 

I - 3 + ») 

Lessons in Horology. 


A'= HV6-a) == 7(^-44 -a) = 7X0.44 
Consequently, if 

7 = 13 one will have N=^ 5.7 turns 

r = H ' N=b.\ '• 

V = 15 " " " J^= 6.6 " 

UL Diameter of the Hub. The custom of making the radius 


! :;! 5! . 

of the hub equal to one-third of the interior radius of the barrel has. 
been established by long practice. When one leaves the hub 
greater than this value, one does not obtain a sufficient number ol 
turns of development of the spring, and when one makes it smaller 
the spring is apt to break, or if it is too much reduced in temper it 

• It H be« not to Bilend tbe figures of the aaann root of 6, becnuse od laooont oribt 
Interior put of the ipHug IhIdk ilUchcd to (he hub, one thereTore Iokb man rmdllr ■ litU* 
of the develDprneot. 


Maintaining or Motive Forces. 65 

Cases present themselves, however, where, in a given 
barrel, one desires to introduce a thinner spring than that which 
one has been in the habit of using and it may then be asked if 
one could not reduce the diameter of the hub, in order to obtain 
a greater development of the spring. Let us then examine this 

It is known that when one submits a rod of steel to a tension, 
the transverse section being equal to one square millimeter, this 
rod lengthens, and that when the load which produces this elonga- 
tion reaches a certain value, a rupture is produced. Now, expe- 
riments made with spring wire have demonstrated that the wire 
breaks if the weight reaches a value of 135 kilogrammes. 

Let us note that the wire which was used for this experiment 

not hardened. 

Let us first seek again the elongation / per unit of length which 
the rod underwent at the instant immediately preceding the rupture. 

We have (88) 



{"aking E= 23000000, P ^ 135000 gr, , L ^ i, we will place 

/ ^ '350O0 ^ 13s 
33000000 23000 

I = 0,0058695 mm. 

On the other hand, let us calculate the elongation of the exterior 

fibre of a barrel spring as it is admitted in practice and compare the 

two results. For this purpose let us take the spring of a watch of 

Tim. (19 lines) which has hjrnished proof of being able to bear 

desired flexion. 

We will have the elongation per unit of length from the 
ula (94) 

!he thickness of this spring is o. 18 mm., consequently 

'In the interior of the barrel and pressed against the drum this 
spring had 13 turns and 5 outside of the barrel ; moreover, after 
beiDg wound, it was set up 5. 5 turns. With this information we find 
■ a = (13 — S + 55) X 2 IT = 37 IT. 

66 Lessons in Horology. 

The length of the spring is 600 mm, ; one will have consequentlf 

Comparing the above figures, 

we can establish the astonishing result that the exterior fibres of a 
spring can sustain an elongation per unit of length twice as great as 
that which produces a rupture by tension. 

This fact cannot be explained by the supposition of a superior 
quality of steel to that of the metal composing the rod which broke 
under the action of a weight of 135 kilog. ; because this last steel 
was certainly of the first quality. It must then be admitted that 
the exterior fibre of a spring does not break, when the elongation 
that it acquires is equal to that which produces rapture by tension, 
for the reason that it is retained by the interior fibres. 

In the presence of this fact one can admit as the limit of 
elongation that the exterior fibre of a spring can bear without 
breaking, the value _ 

One remains within this limit in making the diameter of the hub 
equal to one-third of the interior diameter of the barrel and in using 
a spring making 13 turns in one-third of the interior radius of this 
same barrel. 

When one desires to use a thicker or thinner spring, one must 
in this case determine the diameter of the hub with relation to the 
thickness chosen. Thus the interior diameter of the barrel which 
we have used in the preceding experiment being 17.4 mm., the 
diameter of the hub was then 

iM - 58 mm. 

Dividing this diameter by the thickness of the spring 0.18 one 
arrives at the conclusion that the diameter of the hub should be 32 
dmes the thickness of the spring, in round numbers. If the 
diameter of the hub is smaller than this proportion, the spring runs 
the risk of breaking or, if it is too soft, of setting. 

Work Produced b^ a Spring. 
112. One determines the mechanical work which a spring pro- 
duces at each oscillation of the balance wheel, by dividing the work 

Maintaining or Motive Forces. 67 

displayed by the spring during one turn of the barrel, by the num- 
ber of oscillations made by the balance wheel during this time. 

Let F = 4800 be the moment of the force of the spring of a 
watch whose balance executes 18000 oscillations an hour ; we will 
obtain the mechanical work effected by the spring during one turn 
of the barrel by the product 

H' = 4800 X a w. 

If the barrel of 80 teeth gears into the center pinion of 10 leaves, 
the number of corresponding osciilattona will be 

the mechanical work during one oscillation will then be 


4800 X Jw 
'*'"= ?5x.8ooo -«-»° 

If, on the other hand, the watch only beats i&zoo oscillations 
hour, one would have in this case 

Wm = 

X 16J00 

One sees then that the work of the force received by the balance 
wheel at each oscillation is increased by diminishing the number of 
these oscillations. Let us suppose again that the balance wheel 

I executes 18000 oscillations, but that the pinion of the center wheel 

) has 12 leaves in place of 10, one would then have 

4 S00 y air 

' f5 X 18000 ' 

= 0.351328 gr.ti! 

One, consequently, also increases the force by diminishing the 
f duration of a revolution of the barrel. 

The Fusee 

113. We know that the law of the variations in the action of a 
spring which unwinds, is complex, and that the force developed has 
widely separated limits, for a spring of the same thickness its 
whole length. 

In order to remedy this defect, the ingenious arrangement of 
^e fusee was conceived long ago, consisting of a solid body whose 


Lessons in Horology. 

secdonal revolution is very nearly parabolic, and whose surface" 
grooved with a helicoidal curve. This piece is mounted on the 
axis of a toothed wheel A (Fig. 21) gearing with the pinion of the 
center wheeL The teeth of the barrel are, in this case, suppressed, 
and its arbor remains constandy fixed. 

This arrangement permits of the complete equalization of the 
motive action of the spring. In efTect, when one has just wound 
the watch, the spring is completely coiled ; a steel chain, one end 
of which is hooked to the fusee and the other to the barrel, is at this 
moment almost completely wrapped around the spiral lines of the 
axis of the fusee. On unwinding itself, the spring turns the barrel, 
which communicates its movement to the fusee by the intermediary 
of the chain. This unwraps itself, little by little, from the fusee, 
and wraps around the barrel until there remain no more turns on the 
fusee. It is evident that if the tension of the spring continually 
diminishes, this action works in the contrary sense, always at a 
greater distance from the axis of the fusee. The motive work, 
product of the tension by the distance traversed, gives, designating 
the tension by P, the distance to the axis by r and the speed c& 
rotation by w, 


This work should lje constant if the speed of the wheel fixed 
on the axis of the fusee is constant, that is to say, if there is a 
uniform angular speed, and the fusee is grooved in such a manner' 
that the product P r remains constant. The variations of P must 
then be the reverse of those of r. An exception, however, must be 
made to the preceding, if one takes into account the friction of the 
pivots in the plates ; this friction, in fact, diminishes constantly as 

Maintaining or Motive Forces. 

I tbe pressure diminishes. We will, however, neglect this lactor, in 

■ Brder not to complicate the following theory. 

I 114. To determine practically whether or not the variation of 

■ the force o£ a spring is exactly counterbalanced by the form of the 
fusee, one uses a lever and a weight, as we have seen before (8i) ; 
one fixes the lever on the square of the arbor of the fusee ; the form 
of this piece will be exact if the weight carried by the lever makes 
quilibrium with the force of the spring at the same distance from 
the axis tor each point of the successive rotations of the fusee. 

115. Calculation of the Variable Radius of tbe Fusee's Helix. Let 

it, be the interior radius of the barrel, half the thickness of the 
chain being included therein : 

r, the variable radius of the fusee ; 

r^, its initial radius (in rand r,, is also included half the thickness 

of the chain) ; 
I I, the maximum angle which the spring is wound, starting from the 
position where the elastic effect is null, and corresponding 
to the instant when the chain acts at the extremity of the 
radius r^ ; 

^ the angle which the barrel has turned, starting from the posi- 
tion t ; 

i^ the angle which the fusee has turned, starting from the instant 

when the chain acts at the extremity of the radius r^. 
e moment of the force of the spring can be expressed by 

^ Ee' h 



-- M 

fc will have 

F = M {i — B.). 
The force F' acting at the exterior of the barrel should be 

i the moment F^ with relation to the axis of the fusee is 

The values of r and of a should vary in such a manner that, in 
order that F^ may be constant, we should have a equal to zero for 
'■equal to r^ ; we will then have 


Lessons in 




R R 

(1) «' = 

e — o' 


When the chain wraps up an infinitely small quantity, j? <^ «, 1 
on the barrel, it unwraps the same length, r rf p, from the fusee. 1 
One has then yP rf a = r rf p, J 

but, because of the equation (i), 


On integrating 

it becomes 

9 — n) fl-o. 


These integrals should be taken 
a ^ a', one will have 

between the limits a = o and 1 

To rf a = e a and 

fa ^ a = 5 .' 


Drawing from this equation the value of a. one 

will have first 

- i «■ 4- B a 


changing the signs, adding $■ to ea 
it becomes 

»• - a e a + e' = 


ch member and 

multiplying by 2, 



now in equation (i) a by this valb 

e, we will obtain 1 




Maintaining or Motive Forces. 
Placing fl ^ 2 « M and p :^ 2 « m' we will have, finally, 


116. Namerlcal Calculation of tbc Preceding Equation. Let 

^=8 mm., ''0^5 mm., 8^12X2* and let us calculate first 
the value of the radius r for an angle p ^ 2 ^ ; we will write, on 
replacing values, 

Log : 86 ■- 

- log : 96 -- 



'iveH : 

ax I" 

— 2 

Log; 5 = 0.6989700 

_log:^|^ 0.976.136 


Log : r' = 0.7228564 
and r' — s.aSa;^^. 

Successive calculations will give us in an analogous manner the 
rilowing results, which we group in a table : 

For p = 2 „, 


= 5. 2827. 

" P = A *, 


= 5- 6195- 

" P = 6 -. 


= 6.0302. 

" P = 8 ,, 


= 6,5491. 

" p — 10 w, 


' = 7-2231. 

117. Other Calculations. It more often happens, in practice, 
that one b given the greatest radius of the fusee, and that the ques- 
tion is to determine the variable radius, starting from this value. 
This problem, the inverse of the preceding, is solved in an 
analogous manner. Preserving the same notations as in the pre- 
ceding case, let r^ in this case be the greatest radius of the fusee 
and a the angle which the spring is set up at the instant when the 
a acts at the extremity of the radius r^ of the fusee. We have 
e moment of the force of the spring : 

F^ = M e, 

I in 'he initial position, when the barrel has turned an angle a, 
F=. M(t + a). 

^2 Lessons in Horology, 

The force F* acting at the exterior of the barrel will be for the 
two cases : 

M M 

and the moment of these forces with relation to the axis of the 
fusee will be : 

F'\ = ^^ and /?-'' = ^ (0 + •). 

Making these two values equal, one has 

Mr, ^^ Mr 


from whence one extracts 

As in the preceding case, we place 

r aT p = /? aT a, 
from whence 

ar p = — ar a. 


Replacing r by its value (i), one obtains 

fl^ P = -4 ( e + a) ar a. 

and on integrating, 

from whence 

p - -^- ( e « + i a»). 

Transformations analogous to the preceding case will give us 
successively : 

i a' + e a -- ^ p, 

•« + J e a + 0« = ^ p + e», 

« = - 6 ± ^ 2 ?:^ p + e«, 

Maintaining or Motive Forces, 73 

The value of a, extracted from the equation (i), is equal to 

a = ii - 0. 



from whence 

'•• 6 = ± e ^^l- + X. 

r = 

\ R% 

^ + x; 

or still further, by substituting p = 2 » «' and = 2 v n, 

(2) r = 


+ I. 

118. Numerical Calcalatton of the Preceding: EqnatioiL As an 

example of the application of the preceding calculation, let us 
determine the dimensions of a marine chronometer's fusee and let 
the following be the data : 

Exterior radius of the barrel including half the thickness of 
the chain, R =^ 21.7 mm. 

Maximum radius of the fusee, . . . . r^ = 18.3 mm. 

Development of the spring, « := 3.4 turns. 

Let us admit, that when the spring is set up one turn, the chain 
acts on an angle p == o, in this case then 

n' = o. 

When the fusee has made one turn, we then will have »' = i, 

and on replacing the letters by their values in the formula (2), we 

will have 



2 X 18.3 XI., 

21.7 X 3.4 
The calculation gives 

L(^ : (2 X 18.3) = log : 36.6 = 1.5634811 
—log : (27.7X 3.4) = 1.8679386 

0.6955425 — I 
Corresponding number = 0.4961 

Log : 21.7 = 1.3364597. 
+ log: 3-4 = 0.5314789. 


Lessons in Horology. 



0.4961 + I 

log: 1.496 





Log: 18.3 = 
Number = 


1. 1749708 


will then have the radius of the fusee for a nu 

turns ri 

= I ; 

. = 14.961 mm. 
Replacing, successively, in the preceding formula n' by 2, 3, 4, 

etc, one will arrive at the following results : 

^ 7, 



119. Uniformity of tbe SprmE:*s Force in Fnsee Watcbes. In 
order to obtain perfect uniformity of the spring's force in fusee 
watches, it is not sufficient alone to construct the fusee in a manner 
conformed to the data ol the preceding calculations. There are 
Other factors which must be taken into account, and about which 
we will give some explanations. 

In order to verify the uniformity of the force of the spring with 
rdation to the fusee, or, to speak in shop parlance, in order to 
equalize the fusee, one places the fusee and the barrel between the 
two plates of the watch, puts the chain in place, sets up the spring, 
and fastens on the arbor of the fusee the lever that we have 
DKOtioned (114)- 

Holding the movement of the watch in the hand, one then 
tnnis the lever a quarter of a turn and establishes equilibrium by 
means of weights; then one turns the lever i, 2, 3, etc., turns, 
taking care to notice if at each revolution the equilibrium is 

But practice teaches that if, in this operation, one finds an 
increase of force, one approaches perfect equality by further setting 
tip the mainspring ; if, on the contrary, tbe instrument shows a 

Maintaining c 

'"^ease of force, one lets Ike spring down. When one ass found 
'''s uniformity of tfie force, tlie initial position of the spring's 
'snsion is preserved by marking a point on tfie pivot of the 
"^el arbor, and by repeating this point on the plate of the watch 
9pposite to the position that the former occupies. 

Let us remark that it is necessary to renew this operation each 
le that one replaces the spring experimented with, by a new spring. 

120. Let us seek now for an explanation of the effect whicli is 
(Produced in the preceding experiments. Let us admit that the 
•"adii of the spiral lines of the fusee may have been calculated for a 
Spring which, being wound, is set up lo turns and has unwound 
3 turns at the moment when the chain is completely unrolled from 
the fusee. 

Being completely wound, the moment of the force of the 

Spring is then proportional to lo. 

AVhen the barrel has made i turn, this moment is proportional to 9. 
"When the barrel has made 2 turns, this moment is proportional to 8, 
"When the barrel has made 3 turns, this moment is proportional to 7. 
From the lop to the bottom, the moment of the force has 

diminished ^5 ^= 0.3, 

and in order that equilibrium may be produced, the radii of the 
spiral lines of the fusee must have increased in the same proportion. 
If, on the other hand, the spring was set up only 9 turns, the 
chain being wound on the fusee, we would have in this case : 
Spring completely stretched, moment of the force proportional to 9. 
The barrel has made i turn, moment of the force proportional to S. 
The barrel has made 2 turns, moment of the force proportional to 7. 
The barrel has made 3 turns, moment of the force proportional to 6, 
The moment of the force has then diminished | ^; J = 0.333. 
This decrease is superior to that of the first case, the same fusee will 
not produce equilibrium and one sees thus that it will be necessary, 
in order to have equilibrium, to set up the spring another turn. 

121. We possess in this way a means of regulating, practically, 
the moment of the force of the spring with relation to the axis of 
the fusee, taking into account certain factors which have not been 
introduced into the preceding calculations ; the principal among 
them being friction. 

It is evident that on setting the spring up further, we increase 
its energy ; it is, moreover, only in rare cases that any incon- 
venience will result from this increase of force. 

76 Leaons in Horology. 

122. In establishing the theory of the barrel spring, we have 
admitted these springs to be of the same thickness from one end to 
the other, that their coiled blades remain always concentric, that is 
to say. retain a spiral form during their development ; and, finally, 
that they are always free. 

Practice shows that these conditions are not fulfilled by the 
spring inclosed in a barrel and that they cannot be so except for 
a free spring, such as the hairspring. 

Let us examine rapidly, however, these three facts, commenc- 
ing with the last expressed. 

123. Complete liberty of the blades of the spring does not 
exist practically. Let us suppose, in effect, that one has turned 
the barrel arbof a small angle, a quarter turn, for example. In this 
position all the coils of the spring have not yet entered into play ; 
there are found a certain number which remain pressed against each 
other, forming part of the barrel. One should then take the acting 
length of the spring L, equal to the length of the spring which has 
become free, and the number n equal to the number of turns that 
this free part contains at the instant considered, diminishing this 
value by the number of turns that this same length possesses when 

it is placed freely on the 
table (97). 

Thus (Fig. 22), if a 
is the end of the spring 
hooked to the hub. and 
b the point at which the 
free coils become sepa- 
rated from those which 
remain pressed against 
the drum of the barrel, 
the length L in the 
equation {97) 


is then only equal to the 
*■' length of the part a d tA 

the spring. Furthermore, the number n will be equal to the num- 
ber of coils which this part of the blade contains, diminished by the 
number of coils which this same length contains in the free posi- 
tion of the spring (Fig. 23). For example, we have (Fig. 22) 

Maintaining or Motive Forces. TJ 

fl J = af coils and (Fig. 23) a ^ = 2^ coils ; consequently. 

Let us note that in most watches in which the barrel makes 
three turns in 24 hours, one can readily admit that the whole blade 

^^ becomes free, while the spring is developed these three turns, and 
one can employ, without great error, the above formula, without 
modifying anything therein. 

124. Some have tried to use springs of varying thickness, that 
I, those whose blades increase or diminish in thickness from one end 
D the other. 

I-et us suppose in the first place that the thickness may con- 
stantly increase ; at the interior, the blade, being thinner, will bend 
more easily ; during the winding of the watch, the coils which 
detach themselves from those which remain pressed together, will 
^^Bperform this movement in a more gradual manner than if th^- 

J9 Lessons in Horology. ^^^^^H 

thickness were the same along the whole length. On continuing to 
wind the spring some of the coils will be wrapped around the hub 
and form part of it. Thus, the moment of the force of the spring 
could only be determined in this case by taking the length of its 
iree part alone, and the value of n should also be determined accord- 
ing to this length. Since the thickness, moreover, is variable, it 
becomes difficult to determine by calculation the force of such a 
spring in a sufficiendy exact manner. 

Such springs are used to advantage in fusee watches, because 
they have a more concentric development and consequendy produce 
less friction between the coils. 

If, on account of the diminution of thickness at the interior, 
there results a greater difference between the moments of force of 
the beginning and the end, this difference could easily be corrected 
by the fusee. 

125. Springs thicker at the interior than at the exterior are 
hardly to be recommended, for the interior part of the blade bending 
only with difficulty is hard to wind around the hub ; it has, moreover, 
the efTect of detaching a greater length of blade from the part that 
remains pressed against the barrel, which produces considerable 
friction between the coils. These springs have, moreover, a great 
tendency to break. 

126. The principal defect of the development of the spring in 
the interior of the barrel is that which arises from the eccentric 
coiling or uncoiling of the blades ; these push themselves to one 
side, both at the interior and at the exterior of the spring. This is, 
also, an analogous fact to that which shows itself in a flat spiral 
without return curve. When the interior fault comes into contact 
with the exterior fault, the spring makes a sudden jump, producing 
a noise well known to watchmakers. The exterior fault can be 
remedied by fixing to the spring a flexible check of sufficient length, 
about a half turn ; this check should be made thin in the part 
which is fastened to the barrel, in order to permit it to follow freely 
the coiling up of the spring. A better remedy would be to have 
the last exterior half turn thicker and make it gradually thinner to 
suit the condidons. 


127. We designate by this name a mechanism fastened to 1 
barrel and whose object is to stop the winding before the spring t 
completely coiled around the hub. ' This same mechanism alsoJ 


Maintaining or Motive Forces. 79 

Stops the running of the watch before the spring is completely 
pressed against the inner wall of the barrel ; its eSect then is to 
utilize only a part of the development of the spring, that during 
vhich the force is most equal. Thus the total development of the 
tring being, for example, six turns, ii the unwinding is arrested 
by the stop-work after four turns and a half, 
the spring will still be stretched one turn and 
a half when the watch stops. 

In most watches, in fact, this mechanism 
allows the barrel to make four revolutions 
Fig. 31 on its arbor, and if it makes one turn in 

eight hours, as is often the case, the watch 
should run for 32 hours. 

The most modem stop-work is what is called the ' ' maltese 
cross." It is composed of two pieces, the finger and the wheel.* 
The latter being shaped like a maltese cross, gives it this name. 
The wheel is placed on the barrel, where it 
can turn freely, while the finger is placed 
on the arbor. The head of this piece gears 
in the notches of the cross, the rounded out 
teeth of which can successively slip around 
the drcuniference of the finger. 

On winding the watch, one turns the 
barrel arbor ; the finger participates then 
in thiq movement and pushes, at each turn, ~" "' 

a tooth of the wheel until the moment when the shoulder of the 
filler comes into contact with the full tooth of the wheel ; the 
movement is then stopped, and the watch is wound (Fig. 24). 

During the running of the watch, the 
finger is stationary and the wheel, turning 
with the barrel, at each turn presents one 
of its openings in front of the end of the 
finger, which forces it to make a fraction of 
a revolution on its axis (Fig. 25). After 
flie four revolutions of the barrel, the other shoulder of the finger 
comes again in contact with the full tooth of the wheel, and the 
watch is stopped (Fig. 26). 

128. Geometrical Construction of the Maltese Cross Stop-Work. 

In order to construct, graphically, this stop-work, we will suppose 

*Th«se pieccB lie commoatj knavn ai the male sad fenule — Xs.KS'o^ttH. 


Lessons % 

that the distance O O' between the centers o( the barrel and cS 
the maltese cross wheel 13 known {Fig. 27). We divide this distance 
into five equal parts, and Irom the center O, with a radius equal to 
three of these divisions, one describes a circumference, which 13 
repeated a second time from the point O' as center. From thiS' 
last center we further describe a new circle passing through the 
center of the barrel, and we divide this last circumference into five 


equal parts. In order to get the end of the finger in one of the 
openings of the maltese cross, the division is commenced at the. 
point 1'; for other cases we wilt commence to divide at the point 0\ 
or any other intermediate point. 

The circumference of the finger is described with a radius 
equal to half of the distance O'; from the points i', 2', 3' and 5' 
will trace with the same radius the arcs A C, A' C, etc., the arms 
of the cross. The intersections C A', etc., will, consequendy, 
determine the size of the openings, the straight sides of which 
are drawn parallel to each other and at equal distances from the 
center 0'. 

MaifUaining or Motive Forces. 8i 

The essential conditions to be fulfilled in making the end of 

the finger are the solidity of the piece and the free action of the 

mechanism. We prefer to represent it by means of two arcs of 

circles : one m n^ whose length equals a semi-circumference, and 

the other n ky whose center is found almost on the point of the 

shoulder of the finger. 

Practically, the end of the finger k I should be slightly smaller 
than the corresponding opening of the other piece, that is to say, 
there should exist a certain play, to assure the free action of the 
mechanism ; this play will be easily obtained by taking off the sharp 
comers k and / of the finger. 

It is also to be recommended, in practice, to make the full 
tooth of the maltese cross with a radius O' D longer than the 
radius O' C ol the cut-out teeth, in order to cause the stoppage 
a little before the line of centers. It is, moreover, necessary to 
slightly round ofi the corners A^ Q A\ C, etc., of the teeth of 
the wheel. 




4 dock ha\-« a double dittr to fulfill : firsts to transmit the move- 
ment drtsmg from the motive pover^ from die first mobile down to 
the escapement : seconds to reckon die nomber of osdDations 
accomplbhed by the buLmce whedL in a given time, incficatii^ this 
time by metins ol hjiod^ on a spaced diaL 

Since oa the one hand die movement o£ die fagJance wfaed is 
a rap^i one and on the other die motive fiocce shoold only be 
expended :$Iow{y. and. moreover, die wheelb carrying^ die hands 
^should make certiin munbers ol tumsw accorcfin^ to given rda> 
tioii& one xmdersticcdb that die wheet-^vork slioaki be ai i ai^ed in 
such a manner jb$ to mutti^y. pro^:cessnrd^» die speed of die first 
mobile. Ttbb &$ why we make Ose whedb gieor edCo p^^yrms, and 
die aumber^ of teeth ot diese xffl&tenc msL?bSes slnaki be exsMrtly 

Lee u:< fartS ttfr cesxfiirk tiu£ an dtns coosUaabbr ikMne a Awr the 
^!eed. w^ srnnm&<& st tsue :saaK pco^ioctioii t&e uce tcansoHned to 

irsc ^ae resaDoit wicci <&kxa5£ ess^c ^^ecweest thie amnfcer at 
aoQi&ie :ry nse axcvtn^ S>5fi!? cr a ^^sarin^ ami tiar 
^geoL SJ3cw:n^ nse :tHincer ct 5sea: ^^ it a wa«eL ami <8 
3t rui cmic:tt ^ iat wd»:o: ± <^sa:!Si. we iw^ ^aua: ij; iar :5i^ 

X imiicer %. 

Zjs: xs <arco5e a w^sft^ .^ ^ .3C ^seci ^lescm^: 3t a rnxran m of 
r^ Jta^^*^ snc^ «ic3: 3:^:t3: ^ ^^ wOtteC -irwss »i*^ jstt ^ die 

ijsarr sssTL ^ :ait ««?* wnl ^ur'^ j^Mm^?t£ is iwp:? 

Whtcl- Wart. 


- 33 turns. 

The preceding equation can be presented under the form 

If one wished to know the number of rotations completed by 

the pinion, while the wheel makes any number of them, m ^ 4, for 
instance, the number n' will become w turns greater and one could place 



tfO w 


toiiles I 

The numbers of turns made by the two ; 
aversely proportional to their 
'.bers of teeth. 
131. Let us now consider 
train of gearings formed of 
fO wheels and two pinions 
Fig. 28), the wheel with A 
teeth gearing in the pinion 
of a leaves, on the axis of 
which is fastened a wheel 
with B teeth gearing in a 
pinion with b leaves. 

While the wheel A com- 
pleted n rotations, B made 

The number of turns that the pinion a makes while the 

Iheei A makes a number n, is expressed by the formula (i) 
in the other hand, the number of turns that the pinion b makes 
hile the wheel B makes n' , should be 

t since in this last formula 

one can write, after replacing ra' by its value, 


If, for example, 
the third wheel 10 li 

center wheel has 80 teeth, che pinion 
the third wheel 75 teeth and the 

84 Lessons in Horology. 

fourth pinion 10 leaves, one will find the number of turns that this 
last pinion should make while the center wheel makes i , by replacing 
the letters of the formula (3) by their numerical values, then 

The fourth pinion, then, makes 60 rotations while the center 
wheel makes one. Since the axis of the center wheel carries the 
minute hand and the axis of the fourth wheel carries the second 
hand, the movement will be executed, properly, according to the 
accepted division of time. 

If it were necessary to calculate the number of rotations of the 
fourth pinion, while the center wheel made 24, one would place in 
an analogous manner 

n" = 24 X I -^- " = 1440 tirns. 

132. One could determine, in like manner, for any number ol 
wheels and pinions, the relation of the numbers of turns of the last 
pinion to those of the first wheel. This relation is always equal to 
the quotient obtained by dividing the product of the numbers of 
teeth in the wheels by the product of the number of leaves in the 
pinions. One can then establish, in a general manner 




Suppose, for example, we wish to determine the number of 
revolutions accomplished by an escape pinion while the barrel makes 
4, knowing that this barrel has 96 teeth, the center wheel also 
96, the third wheel 90, the fourth wheel 80, and that all the pinions 
have 12 leaves, except that of the escape, which has 8. We would 
write the formula (5) under the form : 

,,. ,,,, AB CD 

from whence, by replacing values, 

//// -, 96 X 96 X 90 X 80 

133. Calculation of the Number of Osdllatioiis of the Balance^ 

It ia generally customary to indicate the number of oscillations which 
the balance wheel of a watch makes during one hour, that is, while 
renter wheel, which carries the minute hand on its axis, makes 


IVAeeZ-JVork. 85 

134. We have already called attention to the fact that in most 
of the escapements the action of each tooth of the wheel corres- 
ponds to two oscillations of the balance (71). Knowing then the 
number of rotations which the escape wheel makes during one 
hour, one will easily calculate the number of oscillations which the 
balance executes during this same lime, by multiplying the number 
of turns of the escape wheel by twice the number of its teeth, an 
operation which can be represented, designating the number of teeth 
of this last wheel by £ and the number of oscillations by N, by 
the formula 

(7) N^ 2 En"'. 

If an escape wheel with 15 teeth makes, for example, 600 turns 
while the center wheel makes 1, we will obtain the number of oscil- 
ladons made by the balance by 

;V = J X IS X 600 ^ 18000 oscillations. 

135. If we designate by 

B, the number of teeth of the center wheel, 

" " " " third " 
D, •• " " " " " fourth " 
6, " " " leaves " " third pinion, 



E should have, according to the formula (5) 
,„ ._ _ BCD . 

Replacing n"' by this last value in the equaUon (7), we will 
tain the general formula 

a. formula which enables us to calculate the number of oscilladons 
made by a balance wheel during one hour, knowing the numbers of 
teeth of the different mobiles. 

Suppose, for a numerical example, we desire to calculate the 
number of oscillations of a balance, knowing that 

5 = 64 C = 60 D = ba £^15 

Lessons in Horology. 

= iSooo oscillations. 

The application of the formula (8) will give 

jtf- 64X60X60X^X15 

136. Calculations of the Numbers of Teeth. Suppose now we 
wish to calculate the numbers of teeth in the wheels and of leaver 
in the pinions, the numbers of turns or of oscillations being known. 
This question, the reverse of the preceding one, can have severaL. 
solutions ; in short, if one takes the equation J 

t^" _ AB CD I 

' n abed ' ■ 

in which the relation ~ alone may be known, and in which the 
unknown quantities may be A, B, C, D and a, b, c, d, one sees 
immediately that an unlimited number of values could satisfy this 
relation ; the equation is, in fact, indeterminate and affords as many 
unknown quantities as there are wheels and pinions. 

In order to determine them, one chooses arbitrarily the value 
of some of these unknown quantities, and, in order that the result 
will contain no fractions, one chooses for numbers of leaves in the 
pinions those employed in practical use. 

These numbers are, generally, 6, 7 and 8 for the escape piniona,. 
8, 10 and iz for the pinions of the third and fourth wheels, 10, ii_ 
and 14 for the pinions of the center wheels. 

The values a, d, c, d becoming, thus, known quantities, could 
be transposed to the first member of the equation, which will be. 
written under the fonn 

In order to solve this equation, it will suffice, then, to resolve 
all the known numbers of the first member into their prime factors 
and to form these factors into as many groups as there are unknown 
quantities to be determined. 

Let us take a numerical example, and suppose that the relation 
in the above equation be 

-^ = 4800. 
We will then have ^ g rn 

let us choose the following numbers of leaves for the pinions 
a ^ 10, i = 10, c ^ 10 and rf = 7, 
we will place 

Resolving 4800 with its prime factors, we obtain 

4800 = 2« X 3 X 5' 

10 = 2 X 5 
and 7 is already a prime number. 

One will have then the total product : 
3' X 3 X 5' X 7 = 4800 X 'o X 10 X 10 X 7 = 

with the factors of which we 1 
groups : 

1 form the following 

' X 5 = 80 or 

' X 5 = 80 " 

X 5' - 75 " 

X 5 X 7 = 70 " 

etc., etc. 

^ = 2^ X 5* 

£=2» X 3 X 7 
D=2 X 5^ 

By employing other numbers of leaves for the pinions, one can 
niiiltiply the solutions to infinity. As proof, we could have 

4800 = 

3 X 80 X 75 X 7° 
o X 10 X 10 X 7 

^ X Bo X 5° 


137. When the relation ^ is fractional, one factors the nume- 
rator and denominator separately, then cancels the common factors. 
In a case where this elimination could not be efTected, the problem 
would become impossible with the number of leaves chosen, and it 
would be necessary to replace them by others. 
For example, let the formula be 

It is impossible to solve this case with two pinions o 
360 X 1 ° X 1 _ a' X 3' X 5' 

10 leaves, 

lere exists no factor 7 in the numerator which could serve to elimi- 
nate that of the denominator. On the other hand, if one chooses 
the two numbers 14 and 10, one will have 

360 X 14 X 10 _. a' X 3' X 5' X 7 

and will be able to cancel the factor 7. One forms, then, two 
groups with the figures which remain, and obtains, for example. 

Lessons in Horolo^. 

As proof, one will have correctly, 

> determine the r 

.0 X 14 
138. Let the question be, now, I 
teeth in the wheels of a watch, whose balance should make 1 
oscillations per hour. The formula (8) gives 


let us choose for the pinions the following numbers of leaves : 


since the number of teeth in the escape wheel varies only wit 
very narrow limits, we can further replace the letter E by the 
figure 15, lor example. This number is, in fact, that which is very 
generally used for watches of medium size. One will then have 

tax 10X8 ■ 
or, on transposing the known terms, 

> X 15 
Further simplification gives 

16200 X aX4X4 = .ff CD. 
Resolving into prime factors, one obtains 

3' X 3* X 5' = -5 C A 
with which one can form the following groups : 
X 5 = 90 


C = 
D = 

X 3" 
*X 3 
<X 3' 

= 7a. 

The verification of the operation should give 

,6.„ , '° X ;; X 7. X . X .5 . 

We will occupy ourselves, in the problems which follow, witfl.l 
the numbers of teeth to be given to the barrel and to the pinion cfl 
the center wheel. 

139. The number of oscillations of the balance varied gready I 
in the earliest watches ; this figure was governed by no fixed rule I 
and would vary between 17000 and 18000. As these watches had] 

JV&ee/- Work. 89 

no second hands, this number had onlj' a relative importance 
wiliiin these limits. 

In our modern timepieces, especially in watches above la 
lines{27 mm.)i five oscillations per second, which would be 18000 
per hour, are generally adopted. 

In smaller pieces, this figure is increased to six oscillations 
per second, being 21600 per hour, with the object of diminishing 
the influence of jars in carrying, always very perceptible on the 
sinall balance with which these watches are supplied. 

A great many of the English watches beat 16200 oscillations, 
bdi^ 4j^ per second. 

Marine chronometers beat four cecillations per second, or 
1440D per hour. 

140. The problems which follow are the applications of the 
preceding theories and will aid in the better understanding of these 
various questions. We especially insist on the constant use of the 
foraiulas, and urge the pupils to accustom themselves to solve these 
questions by applying to each case the equation which suits it. 
The algebraic way is a sure guide which leads always to a correct 
solution and to exact results. In the following exercises we give 
numerous examples of the reliability of calculation which results 
"^nj the use of the simple formulas which we have just established. 

Problemi Relative to the Preceding Queitioni. 

141. A barrel of So ieeth gears in a center pinion with 10 leaves; 
WW many turns will this pinion make zvhile the barrel makes z 1 

Solution : We have the formula ( i ) which gives 

The pinion executes 8 turns while the barrel makes i ; one revo- 
lution of the barrel has then, in this case, a duration of 8 hours. 
1142, How majiy turns -will this same pinion make ■while the 
'el makes 4. f 
Solution ; The formula (i) further gives 

While the barrel makes 4 turns the center pinion makes 32. 
inre the stop-works are placed in such a manner that the barrel 

ga Lessons in Horology. 

can execute exactly four rotations on its axis (127), the watch will, 
therefore, run for 32 hours, 

143. A center wheel ■wilh 64. teeth gears in a third wheel pinioiK 
■with 8 leaves ; the third wheel with 60 teeth gears in a pinion with 
8 leaves also. How many turns wili this last pinion make during 
one turn of the center wheel ? 

Solution ; The formula (3) 

„_ A B 
" "ad 
gives us, ufter substituting, 

'^ Sirs" ~ ^ 
The fourth pinion will make 60 rotations during one revolution 
of the center wheel, therefore, during one hour. 

144. What is the number of rotations which an escape pinion 
with y leaves will make during 12 koursy knowing that the centeft. 
wheel with 80 teeth gears iji a pinion of the third wheel with to- 
leaves, the wheel of which with J5 teeth gears in the fourth pinion. 
with 10 leaves ; the fourth wheel having 70 teeth ? 

Solution : We will use the general formula for a train of three, 
S«^°S=^ ... ABC 

from whence, after substituting values, 

„/'/ _ ,, V So X 75 X 7° _ 

7300 turns. 


145. What is the number of turns executed fy an escape pinion 
during J turns of the barrel, the wheel-work having the following 
teeth-ranges : 

Barrel .... ^6 teeth. Center pinion 12 leaves, 

Center wheel go " Third " 12 " 

Third " So " Fourth " 10 " 

Fourth " 7^ " Escape " S " 

Solution : Using the formula, we have 

„„ _ A B C D 
" "abed' 


146. Suppose we wish to caladate the number of oscillatums 
which a balance makes during one hour; the center wheel having 
64 teeth, the third wheel 60, the fourth wheel ^6, the escape 

Wheel' Work 91 

wheel i^ ; the pinions of the third and fourth wheels each 8 leaves 
and that of the escapement 7. 

Solution : The formula (8) gives 

,, B C D 2 E 
N = 7 — -^ — , 

from whence ^ ^ ^ 

M = ^X^><f >^^X^^ ^ 14400 oscillations. 

147. What should be the number of teeth in a fourth wheel 
gearing in an escape pinion^ knowing thai the pinion should muke 
10 turns while the wheel makes i ? 

Solution : The equation (2) 

n a 

gives, after replacing n* and n by their values, 

10 _ A 

this equation with two unknown quantities, A and a, is indefinite ; 

several solutions can, therefore, satisfy its demands. Replacing a 

successively by the numbers 6, 7, 8, 10 , we find for A the 

corresponding values, 

60, 70, 80, icx>, , 


10 60 70 80 ICX> 

— = -^ = ■^— = -^ = — = etc. 
I 6 7 8 10 

One obtains, then, the number of teeth in the wheel by multi- 
Paying the number of leaves chosen, by the number of rotations 
^Wch the pinion should make. This is always practicable when 
the number of turns is a whole number. 

If, in place of choosing the number of leaves in the pinion, one 
t^kes the number of teeth in the wheel, the result may easily become 

The equation (2) can be written by making n = i 


a ■= 


Let A = 66, we would have for the preceding case, 

* solution impossible to carry out 



92 Lessons in Horology. 

It is then preferable to choose, always, the number of leaves L 
the pinion, and to determine from these the numbers of teeth in 
the wheels, 

148, To determine ike number of teeth in a third -wheel and th^ 
number of leaves in a fourth pinion combined in such a manner that 
ihe pinion makes 15 rotations while the wheel makes 2. 

Solution : One will use (2) 

and, after substituting 

15 ^ -£ 

Since 15 and 3 are prime to each other and it is, therefore, 
impossible to simplify their relation, it is necessary, in order to 
avoid fractional numbers, that ,f^ be a multiple of 15 and a a 
multiple of 2 ; thus one could have 

The numbers ^5, 75, 90 will, therefore, be suitable for the wheel Ay 
and 6, 10, 12 for the pinion a. 

149. We wish to know the number of teeth in a barrel and /fe' 
number of leaves in the center pinion in which it gears, so thai 
watch may run 8 days with 4 turns of the barrel 

Solution : From the equation (2) we find the value 

in which n' =^ 8 X 24 ^ 192 and « ^ 4 ; then. 

Replacing a by the numbers 6, 7, 8 and 10, successively, one 'J 
finds that 

^ = 48 X 6 =288 
^ = 48 X 7 =336 
v4 = 48 X 8 =384 
^ — 48 X 10 = 480. 

These solutions have the disadvantage of giving too great frl 
number of teeth to the barrel, for even on choosing for a, a pinion 
of 6 leaves, one obtains, still, a barrel with 288 teeth. 

In order to avoid this inconvenience, one sometimes has 
recourse to an intermediate pinion between the barrel and 


Wheel- Work. 93 

center wheel ; the barrel would gear in this pinion and the wheel 
mounted on the axis of this pinion should gear in that of the center 
wheel. The difficulty is thus changed and becomes that of finding 
a place in the watch in which to put another mobile and of increas- 
ing the motive power a very appreciable quantity. 

In order to solve the problem thus arising, we will make use 
of formula (4) 

and we will have 


A B 
~ ab 




A B 
a b 


Choosing for a 12 leaves and for ^10, one has 

^ ^ = 48 X 12 X 10. 

Resolving the two members of the equation into their prime 
factors, one will obtain 

2» X 3' X 5 = -4 ^. 

with which one could form the two groups 

A = 2^ X 3^ = 72 

B = 2^ X 5 = 8<^. 
or else 

A = 2^ X 3 =96 

As proof, one should find that 

12 X 10 12 X 10* 

150. Suppose we wish to determine the numbers of teeth and 
of leaves in the wheels and pinions forming the dial wheels. 
Description of this mechanism. 

The dial wheels are the mechanism whose object is to secure 
the movement of the hour hand. Since the center wheel makes one 
turn per hour, one fixes on the prolongation of its axis, under the 
dial, a second pinion, called the cannon pinion. This adjustment is 
made in such a manner that this cannon pinion participates in the 
movement of the center wheel during the ordinary running of the 
watch, although it is possible to give it a separate movement when 
one wishes to set the hands. Thus the center wheel, the cannon 
pinion and the minute hand have a common movement and make 
one turn per houi. The cannon pinion a gears in the minute wheel 

Lessons i 

A (Fig, 29), which carries a pinion ^ gearing in the hour wheel i 
This last wheel, usually placed on the cannon pinion, around whic 
it can turn freely, is the one which carries the hour hand. Tl 
hour wheel should, therefore, make one turn in 12 hours ; 
wise expressed, the cannon pinion should complete 12 rota 
while the hour wheel nwkis one. 

In the equation (4) one replaces «" by la and n by i, 
unknown quantities are then A B and a 6 ; one, therefore, has 
il = ^ B ^ 

and substituting for a and b the numbers 12 and 10, one will ha\ 

\2X 12X ^0 = A B. 
Resolving into prime factors, it becomes 

=' X 3' X 5 = ^4 J. 
which we can group in the following manner ; 
X 3* - 36 

As proof, c 

X 5 

3 6 X 40 
12 X 10 ■ 

^ 40. 

Wheel' Work. 95 

These figures, 36 for the minute wheel and 40 for the hour 
wheel, are very often employed in practice ; one then gives 12 
ieaves to the cannon pinion and 10 to the minute wheel pinion. 

Evidently other groups can be formed, such as these : 

^ — 2» X 3 — 24 
B 2* X 3 X 5 60, 


A 2* 32 

^ - 3* X 5 - 45, 

or, again. 

A ^ 2 X3X5 — 30 
B 2* X 3 - 48- 

:ie verification always gives : 

24 X 60 32 X 45 30 X 48 

= 12 

12 X 10 12 X 10 12 X 10 

In small watches or low-priced ones, a cannon pinion of 10 

*^^^ves and a minute-wheel pinion of 8 leaves are often used ; this 

drives for the wheels : 

12 X 10 X 8 = ^ i? 

2« X 3 X 5 = ^ ^. 
Trhe two groups ordinarily employed are : 

^ = 2X3X5 = 30 
B =. 2^ = 32. 

Sometimes, also, a cannon pinion of 14 leaves is used and a 
minute-wheel pinion of 8 leaves ; it then becomes 

12 X 14 X 8 = ^ i? 
2« X 3 X 7 = A B, 

from which one can make 

A = 2^ X 7 = 28 
^ = 2* X 3 = 48. 

These last two cases always give 

30 X 3 2 ^ 28 X 48 ^ 
10 X 8 14 X 8 

15L If it were desired to make the dial wheels of a watch 
whose dial was divided into 24 hours, the question would not be 
any more complex, since one would only have to solve the equa- 
tion 2/^ a d = A B, 






(or example 

a = 

2 and i r 

would have 


X 12 


10 = 

: A B 



hence ^^ 

X 3" 


5 = 

= A B, 


= 2* 



= 48 



= 2* 


3 X 

5 = 60. 


ive J 

152. If the same dial ought to show, by means oi two pairs of 
hands of different color or shape, the division of time into 24 hours 
and the division into 12 hours (Fi^. 30), it would be easy to use 

the same cannon pinion and the same minute wheel for both seta 
ol wheels ; one need only add a second pinion, c, fastened on top of 
the first, and gearing in a second hour wheel, C, loosely fitted ( 
the first wheel. 

Admitting for the first train 

Cannon pinion 12 leaves, 
Minute pinion 10 leaves, 

one should have for the first train 

Minute wheel 36 teeth. 
Hour wheel 40 teeth, 

Choosing for c 6 leaves, one will have 

" n X 6' 

from whence 

r - -4 X ■ ■ X 6 . ... 
C j5 4S, 

one will have con'ecdy 


The two minute hands are fastened on the axis of the center 
wted, since in both cases they should execute one turn in ao 
hour ; their angle of divergence once being determined, will remain 

153. Calculation of the numbers of teeth in the wheels of an 
Kironomical clock (_seconds regulator) which should run jj days, 
the weight having a drop of Sjo mm. The cord unwinds from a 
blinder whose radius is /j 7«w. , in which is included half the 
thickness of the cord. This cord is supposed to run through a 

Solution : Since the cord runs through a pulley, it unwinds 
from the cylinder a length equal to twice the descent of the weight, 
'nerefore, i66o mm. On dividing this length by the circumference 
of ihe cylinder, a «■ r, we will obtain the number of turns executed 
l^J this cylinder during the descent of the weight ; therefore, 

I X 3.'4i6 X 15 

: 17.6. 

' The cylinder makes, then, 17.6 turns, while the weight 
(descends 830 mm. ; or, according to the data, during 33 days, 
Wi again, during 24 X 33 ^ 792 hours, 

■ One turn of the cylinder will then be effected in 


: 45 hours. 

Consequently, while the wheel fixed on the arbor of the 
cylinder (fusee wheel) makes one turn, the wheel carrying the 
winuie hand execute 45. 

One sees at once that, in order to avoid having too great a 
number of teeth, one should introduce an intermediate pinion a 
and wheel £ between the fusee and the center wheel (Fig. 31 ). 


98 Lessons in Horolagy. 

In order to determine the numbers of teeth in the fusee a 
the intermediate wheel, as well as for the pinions a and b 
would employ the equation (3) 

FiK- SI 

n which n" = 45 and m ^ i. If we should choose pinion 
8 and 16 leaves, we will place 

■15 X 18 X 16 = >4 ^. 
The first member separated into prime factore gives 
a* X 3* X 5 = -^ -fi. 

Wheel' Work. 99 

^Wch can be grouped in the following manner : 

^ = 22 X 3* = 108 

i? = 2» X 3 X 5 = 120 

We would have, correctly, 

_ 108 X I2Q 
^^~ 18 X i6 • 

To determine the numbers of teeth suitable for the other 

Mobiles, let us note, first, that since the pendulum of this regulator 

^hculd beat one oscillation per second, an escape wheel with 30 

^^^th should execute one turn in a minute (71). One can then 

^^sten the second hand on the prolongation of the axis of its 

Pinion d. The escape wheel executes then 60 turns, while the 

^^nter wheel makes i, and one will have, on employing pinions 

^I 12 and 10 leaves, 60 X 12 X 10 = C A 

"^^liich can give ^ v^ 9 vx 

Z> = 2* X 5 = 80. 

-^s proof, one will have correctly 

60 = 90<_8o 

12 X 10 

154. If^ in place of running jj days, one desired a clock 
^mnning ij months, what should be the numbers of teeth in the 
"^heel'Work, with the same data as that in the preceding problem f 

Solution : Thirteen months calculated at the rate of 30 days is 
^qual to 390 days or 9360 hours. One places 

n^ ^ ABC 
n a b c * 

for one sees that, in order to avoid having wheels with too many 
teeth, a second intermediate wheel must be introduced between the 
fusee and the center wheel. One will then have 

9360 ^ ABC , 
17.6 ab c ' 

that is to say, while the fusee wheel makes 17.6 turns the center 

wheel should make 9360. 

Since the numerical expression 


cannot be employed because of the fraction in the denominator, we 
will transform it by means of the following operation into an 



loo Lessons in Horology. 

equivalent fraction, having as denominator a whole prime 

17.6 " '^' ' 

multiplying this quotient by 11, one obtains the whole numbo' 
5850, then 

936° _ 585^ 

17.6 11 ■ 

One will have, consequently, 

^^ abc = ABC. 

In choosing the numbers of leaves for the pinions, care must 

be taken that 11 is found as factor in one of these numbers, 

order to be able to eliminate the denominator. Let us take, then, 

the figures 22, 16 and 14 ; we will have 

-^ X " X 16 X 14 = ^ -ff C 

3' X 3' X 5' X 7 X II X 13 ^ ^ s C. 

After cancellation, one could £orm the following groups 

^ = 2« X 3 =192 

B ^ 2 X 5 X 13 = 130 

C = 3 X 5 X 7 = 105. 

If one found these values too great, one could choose pinions 

with fewer numbers; for exampie, 10, 11 and 12 leaves, and one 

would have, in an analogous manner, 

^ = 2= X 3 X 5 = 120 
5 = 2 X3 X13-- 78 
C = 3 X 5' =75- 

For both cases we would hate correctly, 

193 X 13° X 105 _ izo X 78 X 7 5 ^ 5850 
22 X 16 X 14 la X II X I" II ■ 

For the other wheels of the train, the case is the same 
the preceding example. 

By introducing a third intermediate wheel, one could succeed 
in making such a clock run for 10 years, but there exist practical 
disadvantages which make this combination seldom used, 

155. How to deiermine the number of teeth in a third wheel 
which has been lost, knowing that the balance should beat 18,' 
oscillations per hour and knowing the numbers of teeth in ike other 
wheels and pinions f 

Wheel' Work, loi 

Solution : Let us call x the unknown number and let the 

Center wheel have 80 teeth, Third wheel pinion 10 leaves 

Third ** ** x ** Fourth ** ** 10 ** 

Fourth ** ** 70 ** Escape ** ** 7 ** 
Escape ** " 15 ** 

The formula (8) admits of placing 

18000= 8oX^X7oXaXi5 . 

10 X 10 X 7 
or, simplifying, 

18000 = 240 X, 

-- 18000 . .. 

X = — -— = 75 teeth. 


The lost third wheel, therefore, had 75 teeth. 

156. Ify in the preceding problem^ the last mobile had been the 
third-wheel pinion^ how would the equation be solved? 
Solution : We would have in an analogous manner : 

,8000 = 80X75X70X2X15^ 

or ;ir X 10 X 7 

- 180000 
18000 = 




10 leaves. 

157. Still using the preceding data, let us suppose that the 
pinion and the escape wheel were both lost, and let us propose to 
determine their teeth ranges. 

Solution : We will have, in this case, two unknown quantities, 
which we will designate by x ^nAy ; the equation (8) will be written 


from whence 


X 10 X jv 


18000 = 

8400 X 




On simplifying. 







The wheel, then, should have 15 teeth and the pinion 7 leaves. 

I02 I-£ssoiis in Horology. 

1S8. In the last problem, we arrived immediately at the ^^ 
numbers ; this does not always happen. Let it be desired, a^S^ | 
second example, to find the numbers of teeth and of leaves i 
center wheel and a third pinion whieh have been lost: 

Solution : One has 

,«™ _ :^ X 75 X 70 X a X tS 

from whence one obtains 

y X 

• X 7 


This result shows us that the center wheel should have eight 
times as many teeth as the pinion has leaves. On replacing succes- 
sively j/ by 6, 7, 8, 10 and 12 leaves, one will obtain the followii^ 
solutions : 





































y = ' 

12 X 1 

Several solutions can, therefore, satisfy the demand, and the one 
which suits best must be chosen ; it is evident, here, that with rela- 
tion to the numbers of teeth in the other mobiles, a center whed 
with 80 teeth and a pinion with 10 leaves are perfectly admissible, 

159. Indicator of the Spring's Development in Fusee Timepieces; 
Murine chronometers and a great many fusee watches carry 2 
auxiliary hand placed on the dial, and with its center on a straight 
line between the point of XII o'clock and the middle of the center 
wheel. The object of this hand is to indicate, on a small dial, the 
number of hours which the chronometer has run since it was last 
wound. It gives notice, thus, of the proper time for rewinding 
the chronometer, an operation which then brings the hand ba(j| 
to zero. 

This mechanism, easy to establish in fusee watches, becomes 
di/licult to introduce in other kinds. Let us take up the first for, 
the present. 

IV/teei - Wori:. 


When one winds one of tliese timepieces, one causes the axis 
oi the fusee to turn, which, once the spring is wound, takes a 
movement in the opposite direction. If, therefore, one places on 
this axis a pinion communicating its movement to the wheel on 
■which the small hand is fastened, the mechanism will be complete. 
The indicating dial is arranged in such a way that the figure XII 
may be wholly preserved ; the hand cannot, tlierefore, make a com- 
plete turn. Let us suppose that while the watch runs 56 hours 
with 8 turns of the fusee, this hand may make seven-eighths of a 
turn and that there thus remains one-eighth of a turn, which is 
taken up by the lower part of the figure XII (Fig. 32). 

The question now, is to determine the numbers of teeth ii 

'^heel carrying the hand a 
^"e fusee, in order to produc' 

given data. 

While the fusee executes r. 

" = J; therefore, {2) 

; pinion fastened on the axis of 
. movement conforming to the 

S turns, the hand should make 


^The number of teeth in the wheel should, therefore, be a mul- 
le of 64 and the number of leaves in the pinion a multiple of 7. 
One can have for 
« ^ 7, j4 =; 64 
a = 14, A = 128, etc. 

When the chronometer is running, the fusee is animated with 
a movement to the left ; the hand, therefore, turns to the right, in 
.the same direction as the other hands. 


I04 Lftsons in Horology. 

It may happea that, owing to the airangemenl of the calibre 
of the wal!ch, one could not make the pinion gear directly in the 
wheel ; it would be necessary in this case to place a second wheel 
{{curing on one side in the hjsee pinion, and on the other in the 
wheel carrying the hand. This last mobile will then take a move- 
ment to the left, and if one wished to avoid that, it would be neces- 
sary to arrange two intermediate wheels between the wheel A and 
the pinion a. 

Let us note that the number of teeth in this or in these inter- 
mediate wheels does not modify the relation existing between the 
movement of the wheel A and that of the pinion a. Designating 
by B the number of teeth in the intermediate wheel, one has 

a D 

160. Simple Calendar Watches. By this name we designate a ' 
certain class of watches having an accessory mechanism by means 
ol which is shown, on the dial of the watch, the date, the day of 
the week and the name ol the month. These indications are made 
by means of hands fixed on the axes ol toothed wheels, performing 
their revolution in the length ol time desired ; that is to say, the 
httnd indiciiting the date jumps each day at midnight, as does also the j 
one indicating the day of the week. These movements are obtained *■ 
by means of a wheel making one rotation in twent^'-four houis^.fl 
TItc dwl showing! the date is ^\ided into thirty-one parts ; the hand ' 
iump*. therefore, for months of thirty-one days from the figure 31 
to the figuiv I. II the month has only twenty-e^ht, twentj'-oine 
or thirty days, the hand should be set to the figure i t^* faa 
Ttns b utconveoient mmI b ovcrcoine in popetoal caletidais, 
Mick iiKclaiu»tts are. thenlore, more conyticated. 

CaleMfer watdMs are often made to incficate also the 
rtwKKKHk Aso(teHing b nUKlein dw dial for this pmpase, 
•irkiMi pan wccw i vrfy representations ol dwnrioas 
wMdi th* WQUK prtse«ts. Tite m w t etj a vk 
W KJMto i a v asittMAiM*^ but iwennittm; ks 
•saKithieMBbcraclKMScI teoJewSv. 

LtH »8 "WTwat wm kwlM ■[»wnr 1 
«Utlt «t feoK* ^ ■rtHtJaa^A ai« eSectad <F^. ssX 

It* MM aafte cQfleBiKtiia* «MsBi 

W^f/- l^ork. 


B and C, of 60 teeth each. These last niobiles execute, therefore, 
one revolution in 24 hours. The wheel B carries a pin m perpen- 
dicular to its plane, which comes into gear at each rotation with the 
^^Ueeth ol a "star" of 31 teeth camnng the calendar hand. This 


last wheel is kept in place by a jump-spring. During the day this 
same pin makes the phase wheel Z. jump in the same manner. This 
wheel, also star-shaped, contains 59 teeth ; it is in like manner 
kept in place by a jumper ; on its surface are represented two 
moons diametrically opposed. When one of the (aces disappears 
behind the dial at the time of the new moon, the edge of the fol- 
lowing is on the point of appearing in the shape of a slight crescent. 
A synodical revolution of the moon (interval comprised be- 
tween two consecutive full moons) is effected in very nearly 2g days 
and a half ; * two lunations require about 59 days. This is the 
reason why 59 teeth are given to the phase wheel. 

• Exuitlr, n iija. IShoun, Mmloutee. 


lessons in Horology. 


The second wheel C also carries a pin, intended to make the 
wheel S, with 7 teeth, jump ; this wheel carries the hand indicating, 

the days of the week. 

The movement ol the hand which indicates the month, is most 
generally effected by setting it by means of an exterior push-piece. 
This hand is carried by a star wheel with 13 teeth, which is kept in 
place during one month by a jump-spring similar to those of the- 
other stars. 

It does not require, therefore, much calculation in order to de- 
termine the toothing of these different wheels. 

161. Suppose it be desired to determine the numbers of teeth. 
and leaves 0/ the wheel-work in a decimal watch, desiring to pre~ 
serve to the balance wheel of this mechanism the same duration of 
oscillation as in that of an ordinary watch. 

Solution : A decimal watch is an instrument dividing the length 
of a day into twenty parts, in place of twenty-four ; therefore, thet 
interval included between a midnight and a midday, or a midday 
and the following midnight, into ten equal parts. Each of these 
"hours" is divided into 100 "minutes." 

Let us further make this a condition : this watch should run 
just as long as an ordinary watch (32 duodecimal hours). An 
ordinary watch, furnished with the customary stop-works, can 
run ij^ days while its barrel makes four turns ; this barrel will 
execute, therefore, 3 turns in a day. In a decimal watch, the 
center wheel should, accordingly, make 20 rotations while the 
barrel makes 3. 

While this barrel makes i turn, the center wheel will make 

- = 6J^ mrns. 

One will, therefore, have 

Choosing a pinion a with 1 2 leaves, a multiple ol the denomina- 
tor of the fraction, it will become 

The barrel would, therefore, have 80 teeth and the center 
pinion 12 leaves (Fig. 34). 

Wheel' Work. 


The fourth wheel should then execute 100 turns while the 
^nter wheel made i ; we will, therefore, have 

B C 

100 =■ 

a b ' 

Choosing pinions of 8 leaves, we have 

100 X 8 X 8 = -ff C. 
l^educing to prime factors, one obtains afterwards 

2« X 5' = ^ C 
^^^ could form the two groups 

2* X 5 = 80 
2* X 5 = 80, 

The center and the third wheel should, therefore, each have 80 

J ^^ and should gear in pinions with 8 leaves. There now remains 

^^ to determine the numbers of teeth in the fourth and escape 


100 turns 1 turn 

< \,w" 

1 turn <— •■ 

1 turn 

Fiff. 34 

^■^^els, as well as the number of leaves in the escape pinion ; 

^^Se numbers have to fulfill the condition declared, not to alter 
"^^ duration of the oscillations of the balance. 

Let us determine, in the first place, the number of oscillations 
^hich the balance would execute during one turn of the fourth 
^heel. In one day this number is equal to 24 times 18,000 ; for 
One turn of the center wheel it should be 20 times less, and for 
one turn of the fourth wheel still 100 times less, which gives 

24 X 18000 ^ .„ .. 

,, — 216 osculations. 

20 X 100 

Let us admit, as is the custom, an escape wheel with 15 teeth. 
The number of turns which this wheel should execute while the 

Lessons in Horology. 

fourth wheel makes one, will be obtained by dividing 216 oscilla- 
tions by twice the number of teeth in the escape wheel, therefore, 

One, therefore, places 

71 -T' 

and, on choosing for the number of leaves a 
example, one will have 

7i X 10 = n. 

D — ■Ji teeth. 

mltiple of 5, 10, for 

which gives 

The fourth wheel could have 72 teeth and the escape pinion 10 leaves. 

162. In the problem with which we have just dealt, the second 
hand will not divide the minute into 100 parts, since it will make 
ai6 little jumps during one revolution. We could still propose to 
divide the minute into 100 equal parts, by abandoning the condition 
stipulated in the first problem, of keeping for the balance the same 
duration of oscillations ; in place, therefore, of making it execute 
2i6 oscillations, let us imagine it as making 200 of them. 

With an escape wheel o! 15 teeth, one arrives at 

^ = 6-/i .«™. 
executed by the escape wheel while the fourth wheel made i. 
Choosing a pinion of 9 leaves, one has 
6?^ X 9 ^ 60 teeth 
for the fourth wheel. 

163. We have still to make the calculation of the wheel-work. J 
for the dial wheels. This problem can have two forms : the one 1 
in which the hour hand should execute i turn a day, and the one.| 
in which it should make 2. 

In the second case, the minute hand will make 10 turns white, J 
the hour hand makes i ; one will, therefore, write 
A B 

with pinions of 12 leaves each, one will have 
10 X 12 X " = ^ S; 

on reducing into prime factors, 

From whence 2' X 3' X S = ^4 ^■ 

^ = 2* X 5 =40 teeth 
£ = 2' X 3' = 36 " 

Wheel' Work. 


If the hour hand should only make one turn a day, one then has 

A B 

20 = 

Taking d 

a b 


10 and a = 8 : 
20 X 10 X 8 

= A By 

« X 5' = ^ ^. 

^°^ could then form the two groups 

A = 
B = 




5 = 40 
5 = 40. 

^^^ minute and hour wheels would each have 40 teeth in this case. 

164. Calculation of the Numbers comprising: the Teeth-ranges 
^' Uie Wheels of a Watch with Independent Second Hand. These 

^^^ches, which were 
^^nstructed in consid- 

^'"^ble numbers some 

^^Bj^ ago, generally 

^^^tained two distinct 

^^ins. In this system 

^ Special hand is placed 

^^ the center of the 

^^^l and makes one 

^^nip only per second ; 

^^ can be arrested for 

^"n indefinite time, then 
^"tcirted again at will, 
"Without stopping the 
Vatch. The office of 
"this second train is to 
tlrive this independent 
second hand. The 
principle of the me- 
chanism is, therefore, 
to release, at each sec- 
ond, the train which 
brings the hand into 
action. For this pur- 
pose the last pinion of 
the second train car- 
ries on its axis an arm p.i^, 35 

no lessons in Horology. 

called the "whip," gearing either directly in the escape pinioi 
the first train or in a "star" adjusted on the axis of this lattec 
(Fig 35)- 

While the whip is in contact with a leaf of the escape pinion, 
it has a slightly-pronounced angular movement, scarcely percept- 
ible on the second hand. But when the leaf of the pinion has 
advanced up to a certain point, the whip becomes free and rapidly 
makes almost a complete turn and again comes in contact with the 
pinion on the nest leaf. At each turn of the whip the se 
hand should advance one division on the dial. At each second, 
therefore, a leal of the pinion or a tooth of the star must present 
itself to receive the whip. 

A lever or a cylinder escapement, whose wheel advances at 
each vibration of the balance, half the space which separates two 
consecutive teeth, can serve for this purpose, ii the number d 
oscillations is 18,000 per hour, therefore, 5 per second. In effect, 
a wheel with 15 teeth produces 30 osdltations and requires, therefore, 

— ^= 6 secoiitls 
to make one turn. 

If one causes the whip to gear directly into the pinion, the 
latter should have 6 leaves ; if not, it would be necessary to fasteni 
a star with 6 teeth, on its axis, into which the whip should be made 
to gear. The movement will then be effected according to the 

There is a remark to be made about watches provided with the 
detent or duplex escapements. 

During the vibration in which the wheel gives the impulse t 
the balance, this wheel advances an angle equal to that which' 
separates two consecutive teeth, and during the succeeding oscilla- 
tion it remains at rest. Owing to this fact each tooth still pro-- 
duces two oscillations ; but we cannot then allow the balance to 
make oscillations, because the whip should become free at 
the end of every five vibrations and, the figure 5 being an odd 
number, there would be found, every two seconds, a vibration 
without an impulse, during which the whip could not be released. 
Watches provided with either system of escapement, should, there^, 
fore, in order to be used as independent seconds, beat an evat 
number of vibrations per second : 14,400 or 21,600 per hour,, 
either 4 or 6 per second. 

IVAeei ■ IVcrk. Ill 

If the watch beats 14,400 vibrations, the escape wheel ad- 
vances two teeth at each second ; the star of the pinion should then 
have -^ teeth ; but, since we cannot have a half tooth, we will give 
15 teeth to this piece, which will amount to the same thing. 

If the watch beats 21,600 vibrations, the escape wheel advances 
3 teeth per second ; the star should have ^ teeth, that is, 5 or a 
multiple of 5. 

One can give to the ordinary train of the watch the numbers 
■of teeth generally employed. Concerning the numbers of teeth in 
the second train, we remark that, since the center wheel carries on 
its prolonged axis the second hand, this wheel should make I turn 
while the whip makes 60 ; one should, therefore, have 

60 = — -r-, 

and employing pinions with 8 and 6 leaves, 

II 60X8X6 = -^^; 

er, reducing into prime factors, 
2' X 3' XS = A B. 
j&ouping these factors, one can have for example. 

A = : 
B = : 

X 3 X . 

X 3 

The other wheels have no other condition to fulfill, except that 
Ae second train should run the same number of hours as the 
ordinary train, generally 32. 

The barrel, which gives motion to the train, has then also stop 
works with 4 teeth, and should make one turn in 8 hours ; that is 
while the wheel carrying the second hand makes 8 X 60 or 
turns. One will have, then, here 
CD E^ 
c'd e ■ 

^loosing pinions of 10, 8 and 8 leaves, one has 
480 X 10 X 8 X 8 = r Z? £■, 
2" X3 X 5' = r/J£, 
nrhtch gives the three groups of factors ; 


1 1 2 lessons i 

165. If the watch has a double set of dials, that is to say, if the 
dial is subdivided into two small dials, the hour and minute hands 
of which can indicate two different times, the pinion gearing in the 
barrel of the independent second train carries a minute hand on the 
extension of its axis, as does that of the center wheel in the trains 
generally used. A set of dial wheels is added to each train, and 
one thus possesses the means of making the watch indicate simulta- 
neously the time of two different countries. In this case the wheel 
which has D teeth should make one turn per hour while the pinion, 
which has e leaves, carrying the second hand, makes 60. The pre- 
ceding figures fulfill precisely this condition, since one has, correctly. 

D E 64 X 60 

166. The arrangement of ■waX.i^^^csXi.^A fifths at quarter seconds 
is similar to that of the independent seconds ; but one could only 
construct such with an escapement whose wheel advances a half 
tooth at each vibration, since the star that is adjusted on the last 
pinion of the second train should become free at each vibration of 
the balance. One could not, therefore, employ in either of these 
systems, detent or duplex escapements. Fifths of seconds watches, 
should beat 18,000 oscillations and the star of the escape wheel 
should present a tooth at each vibration ; this star should, therefore, 
have twice the number of teeth that the escape wheel has. Since 
this last generally has 15, the star should have 30. In place of the 
whip, another star, with 5 teeth, is adjusted on the axis of the last 
pinion of the second train. 

Quarter seconds watches should beat 14,400 vibrations ; the 
star of the escape pinion should have the same number of teeth a 
the wheel, and the star on the last pinion of the second train should 
have 4 teeth. 

The numbers of teeth in the other wheels are the same a 
the independent seconds. 

167. Let us remark that these systems are out of date to-day and 
that they are replaced by the chronographs. These mechanisms are 
simpler and consequently cost less ; they are based on entirely dif- 
ferent principles, having no connection with the kind of problems 
of which we treat now. 

168. Required, to find the number of turns which one should 
give to the winding stem, on setting a watch, to make the minute , 
hand move once round the dial. 

PVAee/ ■ Work, 1 13 

Solution : Given the following numbers of teeth for the wheel 

in action. 

Cannon pinion, 12 leaves Minute wheel, 30 teeth 

Main setting wheel, 27 teeth Small setting wheel, 18 " 
Sliding pinion, 16 " C^'S' 3^)- 

Since it is desired to know the number oi turns which the 
winding stem makes while the cannon pinion makes one, this 
cannon pinion must, therefore, be regarded as the driving wheel. 
The minute wheel, which is driven by the cannon pinion, drives in 

Fig, 8fl 

*ts turn the main setting wheel ; it is, therefore, a pinion with rela- 
*^>Cra to the cannon pinion considered as a wheel and a wheel with 
■~^]ation to the setting wheel considered as a pinion. The same 
*^Wing takes place for the large and small setting wheels, which also 
^i*ive and are driven. One should, therefore, have 


^^B The winding stem mi 

^^Rum, in order that the mi 

^r One sees that the nu; 

l^etween the cannon pinioi 

■^1 the result, and that thi 

la X 30 X ^7 X 18 ^ 11 ^ J_ 
30 X 27 X 13 X iS 16 4 ■ 

St, therefore, be made to execute J of a 
ute hand may make i turn, 
ibers of teeth in the intermediate wheels 
and the shding pinion do not influence at 
/-emcnt takes place as if the sliding 
:, moreover, 

pinion geared directly into the cannon pinion. Wi: hav 
already established this fact when dealing with problem 159. 

169. Let us n(nv seek the number of turns that one should give to 
the winding stem to wind up a watch which has nm a day {2^ hours'). 
Solution : This question deals with the calculadon of the num- 
ber of turns which the winding pinion should make while the ratchet 
fastened on the barrel arbor makes 3. 

Admit the following numbers of teeth ; 

Ratchet wheel 44 teeth 

Crown wheel, lower side, 38 " 

Crown wheel 42 teeth 
Winding pinion 18 " 

114 Lessons in Horology. 

One will have, in this case (Fig. 37), and in s 
to the preceding example 

I analogous manner 

It is generally desired to have this number as large as possible, for 
the reason that the effort which must be made to wind up the main- 
spring, being a determined mechanical work, the lorce which must be 

exerted to wind the watch, will be diminished by increasing the dis- 
tance traversed. One sees that the number n becomes greater when we 
increase the number of teeth in the ratchet wheel and what are called 
the "crown" teeth in the crown wheel, or when we diminish the 
other teeth in the crown wheel and those of the winding pinion. 
169a. Calculation of the Ti^in in a Watch of the Roskopf Type. 
Watches of this kind 

have a simplified train, 

inasmuch as their bar- 
rel gears directly into 
the third wheel. The 
movement of the hands 
is produced by the 
gearing of a wheel A 
(F'g- Sy) concentric 
with the barrel and a 
cannon pinion a placed 
on a tenon fastened at 
the center of the move- 
ment. The wheel A, 
moreover, carries a 
pinion b, gearing in the 
hour wheel B. The 
wheels carrying the hour 

Wheel 'Work. 115 

and minute hands are, therefore, driven directly by the barrel. 
Let us hirther remark that the wheel A and its pinion should be 
adjusted to turn easily on the barrel, in order that the hands can 
be set to the hours. 

169 b. We first propose to calculate the numbers of oscillations 
of the balance in such a watch, the numbers of teeth being known. 

Number of teeth in the barrel C ^ 128 

'' ** ** third wheel . . . Z> = 84 

** ** ** fourth ** . , , E = 60 

'' ** ** escape ** . . . Z' = 15 

<( (i 

= 8 

leaves '* three pinions , . . ^ d = 7 

= 6 

** ** teeth ** minute wheel . . . A =^ 72 

hour '' , . , B = 66 

leaves ** cannon pinion . . a = 18 

minute wheel pinion . d = 22 

The cannon pinion should make one rotation during an hour. 
We will obtain the time of one rotation of the barrel by the quotient 

A _ 72 _ 

T - Is - 4- 

The barrel takes four hours to execute one turn on its axis. 
The number of oscillations accomplished by the balance during one 
turn of the barrel, that is, during four hours, will be expressed by 

the formula 

., CDE2F 

A N = -= , 

c d e 

and during one hour 

.. CD E F 

N = -= . 

2 c d e 

We will have, consequently, 

128 X 84 X 60 X 15 .„ ,. 

^ = — 2 V 8 V 7 V 6 — "^ 14400 oscillations. 

The train of the dial wheels will give, properly, 

72X66 _ 

18 X 22 ~ "• 

169 C Suppose now we wish to calculate the numbers of teeth 
in the train of a Roskopf style of watch, knowing that the balance 
should make 16,200 oscillations per hour. 

ii6 Lessens in Horology, 

Let us admit, as in the preceding case, that the barrel makes 
one turn in four hours. We wiD have 


16200 = J . 

A c d e 

Choosii^ pinions of 8, 7 and 6 leaves, one will have 
16200 X2X8X7X6= CD EF, 
and on reducing the first member into prime Actors, 

2» X 3* X 5* X 7 = CDEF, 
with which we could form the following groups : 


- 2« X 

- 2« X 

- 2* X 

- 2 X 



5 — 
7 — 
5 — 

120 teed 
84 •• 
60 •• 
18 •• 

The train of the dial wheel should give 

A B 

a b 




22 and b 


one has 

A B — 
A B — 





X 18 
X II, 

from whence, 

for example, 

A : 

B = 

2« X 
- 2 X 




: 72 teeth 
66 •• 



170. Definition. The theory of gearings nas for its object the 
study of the transmission of the mechanical work from one wheel 

to another. 

17L Let us suppose, at first, that we have only one wheel 
gearing in a pinion and that in place of the complicated force of 
the spring we have a weight P (Fig. 38) acting through the medium 
of a thin and flexible cord on a 

cylinder whose radius is equal to the . ^. 

unit and which is fastened concen- 
tncally to the axis of the wheel. / 

Let us, at the same time, admit : 
that the resisting force be represented t (^ 

hy a weight Q suspended in the same \ 
manner as P from a cylinder adjusted \ 
on the axis of the pinion and with a '**,.^ 
radius equal to the unit In further i 

imagining this system animated with ^ 

^ uniform movement, the gearing will 
^ perfect if, at no matter what instant 
of the movement, the work of the 
force P is equal and in the contrary 
Erection to the work of the force Qy the relation of the forces 
^ and Q being properly established. 

Since these forces are in the same direction as the path tra- 
versed by their point of appUcation, the mechanical work effected, 
^ Measured by the product of the intensity of these forces by the 
^stance traversed (37). 

If the relation of the forces P and Q is correctly chosen, their 
"^§free may be arbitrary, and, consequently, they can be supposed 
^ Very small or even as nothing. Therein is the basis of the 
^^Portant theory explained in kinetics. 

172. One can also exclude the movement and devote oneself 
^^re especially to the transmission of the force. 

We will examine the gearings from this double point of view. 


Flu. ^8 

Il8 Lessons in Horology. 

173. Practical Examination of a Gcarlne:. Let us place a'D 
and a pinion in a deplhing tool, in such a manner that the two movers 
may be sufficiently free, but witliout play between the points of the 
instrument. Regulate the distance between the two movers until the 
movement of the wheel produces that of the pinion. Impart then 
a rapid movement to the wheel : we will establish a good gearing if 
the movers conserve this motion long enough, and without any othM 
noise than a certain hissing sound easily recognized. The move- 
ment imparted should, moreover, diminish gradually and not 
abruptly. Let us remark that, in order that this experiment may 
succeed properly, the pinion should be furnished with a wheel, per- 
forming the office of a "fly," so that the movement may con- 
tinue long enough. One can also examine a gearing from this 
point of view by placing the movers in the watch and proceeding in 
the same manner. We have thus decided whether or not tha 
gearing transmits the movement properly ; let us now see if it 
transmits the force correcdy. 

Let us use, as in the previous case, the depthing tool, and place 
in the same manner the movers between the arms of the instrument. 
Let us then create a resisting force acting on the pinion, and, for 
this purpose, let us press tightly together the points between which 
the pinion is placed. The gearing will be found established in 
proper conditions if, after imparUng a mo\'emcnt to the wheel, one 
feels no jerks in the transmission and has only the resistance OE 
friction to overcome. 

It is necessary also to assure oneself of the "play" existii^ 
between the teeth of the wheel and the leaves of the pinion and <^ 
the proper space between the points of the teeth and the bottom dt 
the pinion's leaves. 

When the gearing is placed in the watch movement one can 
create a resisting force by pressing the end of a wooden peg against: 
the end of one of the pivots of the pinion ; on causing the wheel to- 
turn with the aid of another peg, one could assure oneself, as in the 
preceding case, of the qualities of the gearing considered. 

1?4. Let us observe that when a gearing transmits the move- 
ment properly, it transmits equally well the force, and when one di 
these conditions is fulfilled the other is, also. It is, however, good, 
for a careful examination, to use the two methods, for certain defects 
make themselves felt more readily by one than by the other of the . 
two modes. 

Gearings, lig 

17S. One will find that for the preceding experiments to indicate 
a good gearing, they must fulfill the three following conditions : 

1st That the distance between the centers of rotation of the 
wheel and pinion must be exact 

2d. That the shape of the teeth and of the leaves must conform 
to theoretical profiles. 

3d. That the total radii of the wheel and of the pinion corres- 
pond to the mathematical calculation. 

We would study separately each of these three conditions, 
Jfhidi summarize all the mechanical theory of gearings. 

lint. — Dutance of the Centeri. 

176. Primitive RadiL Let there be two wheels without teeth 
©and Cy (Fig. 39), one driving the other by simple adhesion and 
■without slipping. 

When the wheel O has turned a certain angle « while driving 

the wheel 0' , the point of contact a, has arrived at b, for example, 

the same point of the wheel O has then reached 6' in such a way that 

arc ab = arc a^ . since the movement is effected without slipping, 

H We can note, 


I O b 

For two wheels having a reciprocal movement, this relation is 
precisely that of the angular velocities (34J : constant when these 
inders have a circular base. Moreover, if the wheel O 

Lessons in Horology. 


has accomplished a number of rotadons c, the wheel (y has ma^ 
number <^ and one would have the new relation, 

177. Although the transmission of mechanical work by ai| 
contact may not be employed in horology, at least in a dl 
manner, one finds, however, numerous applications in the won 
the practical man. In these cases the- wheels are not ordinaril 
contact ; a certain space separates them, and to produce the mi 
ment of driving one by the aid of the other, we wrap around g 
both either a cord, or, perhaps, a leather strap called "the bao 

Thus, for example, the cord of a foot-wheel or hand-wheel in 
watchmaker's lathe transmits the movement, it may be, to a couu 
shaft, or directly to a pulley mounted on the lathe ; the bow st 
transmits, likewise, the mechanical work produced by the hi 
which gives motion to it, to the pulley around which this t 
is wrapped. 1 

1?8. Let us examine, in the first place, the case of two puj 
connected by a cord or band (Fig. 40). Let us first establish 


fact that, if the two sides are not crossed, the two wheels 
the same direction ; if they are crossed (Fig. 41), the wheels 
in contrary directions. 

The angle a corresponding to i turn of the first pull 
equal to 2 » ; for w turns it is 2 ■» fi- 

The same for the second pulley : the angle a' is equal ts 
for I turn and to 2 « n' for n' turns. 

One can then write 


Wheji n, r and r' are known, one has for fif 



If' = II 


and if, as is generally the case, n is equal to i, one has simply 


The number of turns executed by the second pulley while the 
&^t makes i is then equal to the relation between the radii of the 
^0 wheels. 

179. Applications. An arbor makes loo turns to the minute ; 

^furnished with a pulley whose diameter is equal to o.yo m. A 

band transmits its movement to a pulley of 0,4.0 m, diameter placed 

^ a second arbor. One desires to know the number of turns made 

by the second pulley. 

We will have from the preceding relation 


icx) X 070 



n = 100 ; 2 r = 0.70 and 2 r' = 0.40, 

then, performing the calculations, 

nf = 175 turns. 

A pulley of 0.80 m. diameter executes go turns to the minute ^ 
what should be the diameter of the pulley driven^ knotving that U 
should execute 160 turns during the same time f 

The formula 

— = — 

can be just as well written 

2 r 


Lessons in Horology. 

and from thence one establishes 

in figure 

and perlorming the calculations 

2 r' = 0.45 m. 

180. It often happens that a tool, such as a lathe, a counter 

sink, or a drill, should run at different speeds, in order to satisf 
the necessities of the work. One installs then on the driving arb< 
a multiple pulley, tapered pulley or speed cone. On the driven J 
arbor is likewise found a similar pulley, but always in the contrai 
manner. It is only necessary then for one cord to be placed on the 
different pairs of pulleys which correspond. The sum of the radii 
of two corresponding pulleys should then be constant. 

181. Let us now suppose the case of a ioot-lathe. The cord 
of the large wheel is wrapped around the groove of a counter-shaft 
pulley and transcnits the movement to this counter-shaft. Another 
cord is wrapped around another groove of the same counter- shaft, 
but of a different radius, and transmits the movement of the arbor \ 
to the pulley fastened on the lathe. What is the relation between J 
the number of turns of the first wheel and that of the last? 

Let us designate in a general manner 

the nmwber of turns of the large wheel by . , . n 
" " " " " counter-shaft by , , . n' 
" " " " " pulley of the lathe by . n" 

" radius of the large wheel by J^ 

" " " small groove of the counter -shaft by /i' 
" " large " " '■ " " r 

' pulley by . . r" 



We then have (Fig. 42) 

or, replacing n' by its value, 

Since, in this first case, the wheel drives the small pulley 
the counter-shaft, and the large pulley of the counter-shait drives 1 

Gearings. 123 

Ihe small pulley of ihe wheel, one obtains the greatest number of 
turns made by the arbor of the lathe. It is moved, then, with the 

greatest speed. 

li, on the contrary, we 
guide tlie cord of the large 
wheel in the large groove of 
the counter-shaft and the sec- 
ond cord, wrapped in the 
small groove of the counter- 
shait, into the large groove of 
the pulley of the lathe (Fig. 
4j)> we shall obtain a lesser 

Let us remark that, sir 
It IS a mechanical work which | 
should be transmitted, accord- 
"lE as the speed of the last 
pulley diminished, the force 
increases, and reciprocally. 

Thus, when one wishes to 
ton a piece of soft metal, 

such as brass, one arranges the cords in the manner to obtain a 
peat speed, on condition, always, that the object to be turned is of 
small dimensions. On the other hand, if one has a hard piece of 
nietal to turn, such as tempered steel, or an object of large diameter, 
" is proper to arrange the cords in such a manner as to obtain 
Iess speed. 

In the second case (Fig. 43) one has, in an analogous n 
|p the first p p, 

Fig. *« 

182. Numerical ApplicaUon. Let 

a =; I. Ji ^ 400 mm. J?' 30 mm. 

r ^ 50 mm. r' =^ 20 mm. r" 40 mm. 

For the case of greatest speed, one will have 
400 X 5° 
" 30X » 

> as to obtain 

33M turns. 

124 Lessojts in Horology. 

183. The transmission of force by the means of wheels, or 
rolling cylinders driving each other by simple contact, can scarcely 
ever be employed in practice, because the adhesion, called ' ' torce 
ol friction," is very slight; the hmit being passed, slipping is 

To obviate this inconvenience, one inserts in the wheel projec- 
tions, which are the teeth, gearing in the openings contrived in the 
pinion. One then forms 
what has been called the 
leaves of the pinion. With 
this arrangement the move- 
ment of the two toothed 
wheels should be made in 
an identical manner to that 
of the cylinders first con- 

It, therefore, follows 
that in a gearing one can 
always imagine two circum- 
ferences driving each other 
by simple contact, and ia 
the same conditions oE 
movement. These circum- 
ferences bear the name of 
primitive circumferences. 

184. One calls the ^;Vi-A 
of the gearing the length 
of the arc measured on the primitive circumference of one of the 
wheels, extending from a point of one tooth to the similar point of 
the tooth which follows. The pitch of the gearing should then 
comprehend the space occupied by a whole and a blank of a tooth. 
The pitch of the gearing of the wheel should be equal to that 
of the pinion which it drives. Let us designate this pitch by the 
letter^ and call, moreover, the number of teeth in the wheel «, and 
the number of leaves in the pinion n'. 

The length of the primitive circumference of the wheel, a w r, 
should then be equal to ^ X «, since the pitch ought to be con- 
tained n times in this circumference. 

For the same reason the length of the primitive circumference 
of the pinion, 2 ' r', should be equal to^ n'. 


Gearings, 125 

In order to obtain a relation between the primitive radii and 
the numbers of teeth, let us divide the equation 

2 wr = / « 

2 w r^ = / «' 
we will obtain 

2 t t r / « . 

tT'k'P ~" p n' ' 
or, after simplifying 

(3) -;7 = li;- 

Tk primitive radii are then proportionate to the numbers of teeth. 
185. Calculation of the Primitive RadiL In an exterior gearing, 
such as that which we have considered (Fig. 38), the distance 
between the centers of the two movers is equal to the sum of their 
primitive radii ; that is to say, one should have 

(4) D^^r^-r^, 
^ representing this distance. 

Let us take up again the proportion (3) 

r n 

^ which the radii r and r' are unknown quantities and the number 
of teeth n and n' known quantities. 

Without changing the value of an equation, one can add to 
each of its members the same term, or an equivalent term. We 
can then write 

r" '^ r^ ~ n' ^ n'' 

since the two terms -^ and -— - are both equal to i. 
The common denominator permits us to write 

r -\- r* n -\- nf 



and because of (4) one y 

ivill also have 

from whence we deduce 


n -^ n' 


r^ — 

D *^ 


In an analogous manner we would find 


n « 

n + n^' 


Lessons in Horology. 

186. rJumerical Application. A barrel ol 80 teeth should g( 
in a pinion with 10 leaves, what should be the primitive radii -^i 
the two movers, knowing that the distance tietween their cent& jxs 
is 11.565 mm.? 

Replacing in formulas (5) and (6) the letters by thi 
above given, one will have 

t" = 11.565 X s 

1-565 X 1 


+ 1.385 - 


vain -^ei 


These two calculations give 

7' ^ 1.2 

r = 10.2 

As a verification, one should have 
D = r + r' = 10.28 

187. To obtain the primitive radii, one can also simply regar"-" 
the distance D as divided into as many parts as there are teeth i^^ 
the wheel and the pinion together ; therefore, into n -)- n' parts:^ 
and appropriate a number n of these parts as the radius of th^* 
wheel and a number n' for that of the pinion. The calculation i^^ 
thus brought back to that of the preceding example. 

188. The case of exterior gearing is the one which is most gen — 

erally presented in practice. 
In this system we will observe 
that the movement oi the 
two mobiles takes place ii* 
contrary directions ; whoi ' 
the wheel is animated witltfl 
a movement to the right, I 
the pinion will possess a I 
movement to the left* I 

189, When the center % 
of rotation of the pinion is 
placed in the interior of 
the wheel' s circumference 
(Fig. 44), the gearings thus 
constructed take the name 1 

. thus J 
name ■ 



o( inlerior gearings. In this case the pinion takes a movement in 
the same direction as that of the wheel. 

The distance between the centers is then equal to the difference 
between the primitive radii of the two wheels. Therefore, 
(7) ZJ = f- - f^. 
If the distance between the centers and the numbers of teeth 
in the wheel and pinion are known, the value of their primitive 
radii can be calculated in an analogous manner to that which we 
liave just employed to determine those of exterior gearings. We 
have the proportion (3), 

"fiich can be written 

^ r^ _ n n^. 

r' t' ~ n' Jt' ' 

O"". again, 

r — r' _ n — fif 
r' «' ' 

on replacing r — r" by its value D, 
D _ a — n' 
r' - V ■ 
"Om whence we find 

(8) r' =^ D ^^^^ 
'R an analogous manner one would arrive at the conclusion 


^ n ' 

190, Numerical Application. Let us take as a numerical ex- 
ample that of a wheel with ijo teeth gearing interiorly in a pinion 
with 14 leaves, the distance between the centers being 8.75 mm. 

The application of the formulas (8) and (9) give : 
_ 8.75 X 14 

- = 8.7s 


performing the calculations, one arrives at the conclusion 

: 1.156 mm. 
: 9.906 " 

Tile verification should ; 

I ways give 


1 38 

Lessons in Horology. 

I9L Let US now examine a kind of gearing sometimes employed 
and which is called rack gearing. In this case the primitive circum- 
ference of the wheel becomes a straight line ; its radius is, conse- 
quently, infinite and the number of its teeth unlimited. This gear- 
ing can be considered either as exterior or as interior, for the dis- 
tance between the centers can, equally, be 

D = o) + r' = a)— r' = co. 

To determine the primitive radius of the pinion gearing in the 
rack, it is sufficient for us to know the number of its teeth and the 
pitch of the gearing. 

In Fig. 45 let a b equal the pitch of one of these gearings, 
and place , 

let us call n' the number of leaves which the pinion should have ; 
the primitive circumference will then have for its value 

a w r' = v4 «', 
which gives 

(lo) r* = ^^^ 

192. Numerical example. Let 2.8 mm. be the pitch of a rack 
gearing, the pinion must have 12 leaves, what should be its primi- 
tive radius? 

The formula (10) gives 

" ^ ^ 5.347 mm. 

A n' 2,8 X 13 

=.8X 6 

2 IT 2 X 3-1416 


The radius sought should then be 

r' = 5.347 


Gearings. 129 

193. Application of the Theory of Primitive Radii to the 
Escapements. The theory of gearings finds its application not 
only in the wheel-work, but also every time that there is a question 
of tile transmission of movements of rotation around two fixed axes. 
It can then be applied also in special cases, such as one encounters in 
the study of the escapements, the mechanisms of repeaters, etc. 

It sometimes happens, and especially in the last cases, that one 
knows the distance between the centers of rotation and the relation 
of the angles traversed in the same time by the movers considered, 
and that one may have to determine their primitive radii with the 
object of finding out the form of the surfaces in contact. 

The formula ( i ) gives us the proportion 

■which indicates that the primitive radii are inversely proportionate 
to the angles traversed in any equal times. 

Furthermore, one should have, when the rotations of the two 
movers take place in opposite directions, 
D ^r ^ r', 
and when they lake place in the same direction 
D = r — r". 
On performing identical operations to those which we have 
indicated (185), one will arrive at the following results ; 

I Ex ten or gearing, 


Interior gearing, 


(14) r'^n -r~r,. 

194. Numerical Example. To find the primitive radii of an 
escape wheel and of the anchor, knowing, that while the wheel 
traversed an ange of 10° -- a, the anchor turns an angle of 9° =^ a!. 
Moreover, let the distance between the centers ht D =-- 100 mm. 

Let us remark that, the wheel being animated with a move- 
. ment to the right, the anchor possesses a movement to the left, 


Lessons > 


when the tooth acts on the exit pallet, and a movement to the right 
when it acts on the entrance pallet. The first case is, then, that 
of an exterior gearing, while the second is similar to that of an 
interior gearing. 

The formulas (ii) and (12) will give us 

r = D — -— = 100 — ^ = 100 - 
a -}- a' 10 + 9 I 


~ d 4- a' ~ '°° 10 + 9 ~ '"* 1 

The formulas (13) and (14) wilt then give i. 

r = D ^^ _ ,,» ^'^ _ .00 > 


Second— Form of the Teeth u)d Leave*. 

195. General Study of the Transmission of Force in Gearings. 

In the chapter on motive forces, we compared the energy displayed 
by a motive spring to the effect produced by a weight placed at 
the extremity of a lever arm equal to the unit of distance, the 
system being in equilibrium (83). 

This fictitious weight F represents the moment of the force 
with relation to the axis around which this force exerts its action. 

By means of the gearing, this action is transmitted to the 
second axis and the problem is to find the moment F' of a force 
which, with relation to the second axis, would be in equilibrium 
with the moment F. 

196. Let us suppose at first that the point of contact of the 
tooth of the wheel with the pinion leaf is found on the line erf 
centers (Fig. 46), and regard the wheel as a lever in the state 
of equilibrium. This system fulfills in effect all the conditions rela- 
tive to the lever ; the fulcrum is (9, tiie power is F; the resistance 
is that which arises from the wheel (7 and the moment of which 
we have to find. 

This resistence is applied at the point of contact, c, of the wheel- 
tooth and the pinion-leaf ; it is directed normally to the surfaces in 
contact ; here, perpendicularly to the line of centers and consequently 
following c N. It acts thus in a contrary direction to the force F. 



The lever arm (43) of the force N \s O c =^ r^ its moment 
is then ^ ^ ^ 

and because of the equilibrium, one should have (43) : 

(15) F=Nr, 

since the lever arm of the force F is equal to the unit 

On the other hand, the pinion is acted upon by two forces : 
one, F\ is the resisting moment to be determined ; the other, N'y 





Fig. 46 

coming from the tooth of the wheel O and acting, as also does the 
force Ny in the direction of the common normal at the point of 

Since the pinion, as well as the wheel, is in the state of equi- 
librium, one should have, in an analogous manner, the equality of 

the moments : ,,v -,, Ar>.y ^ 

(16) F^ = NX r^. 

On dividing the equations (15) and (16) member by member, 

^ Nr 

F' '' 

one has 

' N' r^* 

The normal forces JV and iV' are equal, since their effects 
destroy each other ; consequentiy, one obtains simply 



132 Lessons in Horology. 

from whence one finds the value sought 

(i8) F' = F~. 
On account of the propordoa (3) : 

196a. If, for example, the moment of F is equal to 4000 gr., 
the number of teeth in the wheel n :^ 80 teeth and the number of 
leaves in the pinion n' = 10 leaves, the formula (19) would become 

f = 4000 8^ = 500 gr, 

A weight of 500 gr, suspended at the extremity of a lever 
arm i mm, from the center of the pinion would then make equi- 
librium with a weight of 400a suspended at the same distance 
from the center of the wheel. 

Let us remark at this time that if the force has diminished 
during its transmission, and is not more, with relation to the 
pinion, than the eighth part of what it wag with relation to the 
wheel, the speed of the last mover is, on the other hand, increased 
and has become eight times greater. 

197. Supposing that the preceding calculation relates to the 
gearing of a barrel with the center pinion, let us now seek for the 
moment F" of the force that should be applied to the third wheel to 
muke equilibrium with the moment of the force of the barrel spring. 

We have seen, in the preceding case, that on multiplying the 
moment /" by the relation -", one obtains the moment of the force 
applied to the center wheel ; on multiplying, then, this latter value 
by the relation -^-, of the number of leaves in the third wheel 
pinion to the number of teeth in the center wheel, one will obtain 
the value sought, thus : 


198. One could continue this reasoning for any number of 
wheels. Thus, the moment /""" that should be applied to the 
escape wheel to make equilibrium with the moment of the force 
of the spring, will be e.xpressed by 

_ n' tt" 

l»i) /^ = 

/■- - — 




199. Let us choose as numerical example the very frequent case, 

The force has become 4800 times weaker but the speed of the 
last mover is 4800 times greater. That which, in mechanics, is 
lost in force is gained in speed and reciprocally. 

200. We have just studied the transmission of the moment of 
the force from one wheel to another, admitting that the point of 
contact of the movers is on the line of centers. 

Let us now see under what condition this point of contact can 
be found outside of that line, in such a manner that the moment of 

Fiif. 47 

force transmitted preserves at each instant the same value that it 

possessed when the contact took place on the line of centers. 

Otherwise expressed, the question is to form the teeth and the 

leaves in such a manner that the^ transmission of the force may be 

constant. It is necessary, therefore, that the value given by the 

formula (19) 

F' = F 



remains the same no matter what the position of the movement. 

134 LessoTU in Horology. 

201. Let us suppose that the wheel-tooth and the pinion-leaj 
are formed in such a manner that at one instant of movement ttiis 
contact is found at the point c (Fig. 47), situated outsids of the 
line of centers. Let us find, in this position, what would be the 
value of the weight F' which would make equilibrium with thc' 
weight F, these two forces being placed at the unit of distance 
from the axes. 

The normal to the point c along which is exercised the 
reciprocal action of the tooth on the leaf and the leaf on the 
tooth, is necessarily normal both to the curve of the tooth and 
the form of the leaf, since these two lines are tangent at this point; 
it is directed along the straight line N N'. 

As in the preceding case, the two wheels can be compared to 
levers. The wheel O is, in effect, acted upon by two forces. The 
one, F, tending to impart to it a movement to the left ; its normal 
is /^ X I, therefore, F\ the other, A'^ directed in the opposite 
direction and arising from the pinion leaf, its lever arm being the 
perpendicular b, its moment is 

.V X Ob. 
Because of the equilibrium, one will have (43) 
(22) F^ NY. Ob. 

The pinion is likewise acted upon by two forces : the one, F\ 
whose moment is F' ; the other, arising from the pressure that the' 
tooth exerts on the leaf at the point c, following the normal direc- 
tion c N' ; it moment is 

N' X O' b'. 

Since the direction of this last force is inverse to that of F', the 
equilibrium is produced by the equality of the moments : 

{33) F' = N' X O' I/. 
Dividing equation (22) by (23), one has 

Since equilibrium exists in the system, the forces JVand .A", 
which have the same alignment must be equal ; in consequence, one 
has, after simplifying, 

Gearings. 135 

The two triangles 6 a and O' U a are similar ; their homolo- 
I plus sides give ihe proportion 

but since (3) 

~V ^ ~^' 
one will also have 

O b _ n_ 
(y b' n" 


F _ n 
F' "»'"' 
'roin whence one finds the value 

^F' ^ F ~. 
202. The value of F', identical to that which we have deter- 
''^'Hed in the preceding case, is then realized, and the iorce traas- 
"*ltted from one wheel to another will remain constant, if the 
"*^l"mal common to the point of contact of the tooth and of the 
'^f passes, in no matter what position of the movement, through 
"^^e point of tangency of the primitive circumferences. 

203. To recapitulate, we can deduce from the preceding 
demonstrations the following rule, which is the basis for the de- 
termination of the forms of contact of teeth and leaves. 

In order Ikai Ike transmission of force by gearings may 
remain conslani, it is necessary tkat ihe acting surfaces of the 
teeth-ranges be formed by suck curves that at any instant of the 
movement tke normal common to the point of contact passes always 
through the same point of tke line of centers, which is ihe point of 
tangency of the primitive circumferences. 

204. It follows from this law that when the contact takes place 
on the line of centers, this point is blended with the point of 
tangency of the primitive circumferences. 

205. Let us remark that, if the normal cuts the straight 
hne O 0' between the points O and (7, the gearing is exterior 
and the movements of the two mobiles take place in opposite 

If the normal cuts the straight line O C outside of the points 
and 0\ the gearing is interior, and the movement of the two 
wheels takes place in the same direction. 


Lessens in Horology. 

If the normal cuts the line O O* at the point Cy, the radius ^ 
becomes nothing and one has 

P' = F~ = O. 
the transmission of the movement of the force is impossible. 

If, on the contrary, the norma! cuts the line O (7 at the point 
O, one has in this case r ^ o and consequently : 
F' = F— = <xi\ 

the force F' becomes infinitely great, but the transmission of the 
movement is wholly impossible, since the primitive radius of the 
wheel is annulled. 

If, finally, the normal was parallel to the line O U, one would 
then have f' ^ F — = F 

This could be the case with the entrance pallet of the anchor escape- 
ment if the escape wheel should traverse the same angle a as the 
anchor which it drives ; one has thus (193) : 

the primitive radii are then infinite, 

206. The law which we have formulated (203) shows us, even 
from the beginning, that the problem whose object is to find the 
curves of the teeth and leaves is susceptible of a great variety of 
solutions, for one may give to the teeth of one of the wheels any 
special form and find such a curve for the teeth of another wheel as 
should satisfy it, in its successive contacts with the first, according 
to the conditions given. However, the laws of the resistance of the 
materials, the wear of the rubbing surfaces, the inflexions of the J 
curves, are so many causes which make us, in practice, reject thw 
use ol a number of these solutions. 

207. Let us further remark that the formula F' := F ^ is < 
independent of the absolute value of the primitive radii r and r" * 
and depends, consequentiy, only on the relation of their primitive J 

Determinatioii of the Formi of Contact in G«tirin^. 

208. There are several methods serving to determine the bear^fl 
ing surfaces of teeth and leaves ; the basis of these different con- 
structions rests generally on the law which we have set forth C203)J 
We will study here three of the principal of these. 

Gearings. 137 

209. First— Graphic Mctliotl. Exterior Gearing. The funda- 
mental condition, that the common normal to the point of contact 
oi two forms which drive each other should invariably pass through 

the point of tangency of the primitive circumferences, himishes an 
easy graphical means to determine one of the curves, when the 
other is given. 

Let and C (Fig. 48) be the primitive circumferences of a 
iring and A B the given curve of the pinion in any position. 

138 lessons in Horology. 

If from the point of tangency a we draw a normal to t 
curve, we will thus have the point of contact i of the leaf of t 
pinion and the tooth of the wheel corresponding to the posida) 

Let us remark that, in this position, the normal a i forms t 
same angle with the radius r' of the pinion as it does with the 
longation of the radius r of the wheel, since these two lines r 
into each other. 

Let us afterwards mark on each of the primitive circumference 
a point, b and ^, determined in such a manner that one may hav^ 

Through the points 6 and b' draw the radii O b and ff 6', pr( 
longing the first suiSciendy beyond the circumference of the wheel 
from the point b' trace the normal to the curve b' 2', then lay bl 
from the point b as summit, an angle equal to 2' b' C and mark thi 
point 2 making b 2 equal to b' a'. The point 2 belongs to the 
curve sought, for if the points b and b' arrive at the position a, the 
radii O b' and O b will have the same alignment and the points 2 
and 2' the same position. 

One can thus determine as many points as one wishes, and, 
connecting them by a continuous cur\'e, one will obtain a form sui 
as /? C, possessing the ability to drive the curve A B \a such a 
manner that the transmission of the movement may be uniform. 

If one conducts the curve A B in such a way that the point Af, 
which belongs both to this curve and to the primitive circumference 
of the pinion, presents itself at the place of the point a, the point A^ 
which belongs to the curve sought and to the primitive circumference, 
of the wheel, should enter into contact with the point Af. 

Thence it follows that one has 

and also that when the contact takes place on the line of centers it 
is found at the point of tangency of the primitive circumferences. 

210. Interior Gearings. For an interior gearing, one deter- 
mines the curve of contact in the same manner as for an exterior 

One describes the primitive circumferences O and O" tangent 
to the point a (Fig- 49) and the curve given A B, which we will 
suppose anew to be that of the pinion. On drawing irom the point a 



Gearings. 139 

fiie normal to the curve, one detennines the point of contact i cor- 
responding to the position given. 

Let us indicate afterwards on the two circumferences the equal 
area a b and a 1/ , a c and a (', etc., laying off from the points 
b, c, etc. , angles equal to the angles that the normals // 2' , tf 3', etc. , 
iorm with the radii 1/ O , ^ (y , etc. Afterwards making b 2^^ 1/ 2', 

c 3 ^ f' 3', etc. , we determine the points 2, 3, etc. , belonging to the 
curve sought. The only difference between this drawing and the 
preceding one lies in the fact that for the exterior gearing, one lays 
oil the angles z' b' O', 3' f' (/, etc., on the prolongation of the radii 
ob, Of. .etc., of the wheel, while for the interior gearing one lays 
these angles off from the radii themselves. 

Since we can choose arbitrarily one of the two curves and seek for 
the Other, we can see that the problem allows an infinite number of 

solutions ; let us remark, however, that a number among ihem preal 
inconveniences, and even impossibilities, for practical execution. S 

211. Second— Method of the Envelopes. The centers of rd 
tion of the two wheels are habitually fixed and the mobiles 
around these points. 

Let us suppose, however, that a movement of rotation 
have been imparted to the whole system around one of the cenl 
that of the wheel, for example, and that this movement is execu^ 
in such a manner that its angular speed may be equal to, but fel 
contrary direction to the angular speed animating the wheel O. 
is evident that from this method the wheel remains in a state i 
repose and that the working of this gearing will remain the sait 
as if the two centers were fixed and the two wheels turned sini^ 
around their respective centers. 1 

The gearing of the fourth wheel with the escape pinion | 
timepieces called " tourbillon " offers an example of such a mov 
ment. The wheel is screwed on to the plate of the watch ; 
movement is, therefore, null with relation to this plate. The esca 
pinion, pivoted in a mobile cage, turns around its center and simi 
taneously with the cage, whose center of rotation is also the cenl 
of the fourth wheel. 

The principle of the method of the envelopes r^ts on this w 
of movement. J 

Let us adopt, in short, any form of leaf ; on representing 1 
pinion in several successive positions of its movement, arott 
the wheel, we will obtain the form of the tooth, on joining by 
tangent curve the positions that the leaf will occupy during tl 

One can then say that the tooth is the "envelope" ofthediffert 
positions occupied successively by the leaf during the movement 
the pinion around the center of the wheel. With this method, t! 
tooth remains constandy in contact with the leaf, and the transm 
sion of the force will be effected without loss. The movement 
the wheel being uniform, that of the pinion will consequently al 
become so. U 

Let us take some examples : l| 

212. The transverse section of the pinion leaf of a /anU 
^■earing is a circle whose center is situated on the priimti 
circumference. Suppose we wish to determine the form of t 


If (Fig. 50) the circumference passing through the points 
'"1 2", 3", ... is the primitive drcumference of the wheel. 

(y A B that of the pinion, and if this last is moved without 
)ing, around the primitive circumference of the wheel, the 

Lessons in Horology. 

successive centers of the pinion will occupy in turn the points 
o', i', 2', 3', . . . On conceiving, then, the corresponding positions 
of the pinion leaf whose centers should occupy the positions 
o, I, a, 3, . . , and on drawing the curve abed,... tangent to 
the leaf in these several po- 
sitions, one will obtain the 
curve of the tooth sought. 

One sees that. 
pinion leaf is reduced to a 
point, the form of the tooth 
would be an "epicycloid" 
whose generating circle 
would be the primitive circle 
of the pinion. If the leaf 
is formed by a cylindrical 
pin, the curve for the tooth 
which results from it is par- 
allel to this epicycloid, and is 
found removed a distance, 
equal to the radius of the pin, 

■ i One could draw a sec- 

^^^^^^^^T ^i ond curve tangent exteriorly 

^^^^^^^^f to the several positions that 

^^^^^^^H I the pin occupies during the 

^^^^^^^H movement the curve thus 

^^^^^^^H formed would then drive the 

^^^^^^^H pin by its concavity. 

^^^^^^^H The straight lines i" 

^^^^^^^H ^1., which connect the 

^^^^^^^^H (*' ^ points of tangency of the 

^^^^^^^^H primitive circles with the 

^^^^^^^^V : p point describing the epicy- 

^^^^^^^^ Fie 01 cloid are normal to the curve 

^ at these points. 

^1 213. Fig. 51 shows that if the contact of the tooth with the 

^f leaf took place at a point such as c, before the passage of the 

line of centers, there would be produced an abutting which, 
if it did not absolutely prevent transmission of the movement, 
« would, however, modify considerably the uniform transmission 
^■^ of the force. 



One knows, in fact, that the normal to the point of contact 
should pass through the point of tangency t of the primitive circum- 
ferences; but one recognizes that this essential condition is not 
fulfilled in this case, since the normal cuts the line of centers at a 
point b. In place then of being (201) 

the moment of the force transmitted will no longer be expressed 
ejtcept by the value 

^ Ob' 

friction being left out. One recognizes, thus, that in lantern gear- 
'"Ers, the contact of the tooth and of the leaf should commence 
^ery near the line of centers.* 

Note. — If the normal passed through the center ff of the 
fnion, the movement would become impossible ; il it passed on 
"'e other side of O", the pinion would turn in the opposite direction 

^JScily It the ) 
°^ tb/iooth tur 

i„2P« hetween Iha line of eent 
iPaeilou. IhB beBinnlPg "f the I 

pin conimences. Let X (Fig. 
"^asiIonBt thin Instant. The 
=«toiooD to tha tooth gnd to Iba jiln, pisws IShiubIi 

"eaia O of the plo. Tha dlsiance O J is at once llie 
rMios of (ho pfn and thB radiu* of curvaturo of the 
•Moreloid JV Oat IbB point O; let u>, Ihetefore, dosig- 

il at Ibirpoii 

tute it bj 6. The diitanca A la that wbic 

h, iQ the 

bnaalt cited Id the teit, 

- B Jf 

8-" + S'-TTb 

Ij dpsinTialed by n, and the dlslaoM A X. 

equal to 

B — n, ia very nearly the diaUnce sought, be 

iDRle of XO with thfl line nf centern diffcra v 

erj Utile 

ftcSn a right angle (tf it diffeta -eo.lblj In the 


Il becmuM the radius of the pin fj Is enag^ 
prevanl fho confiiiloo of the lines). But th 

rated to 

J JT Bought to the radius 8 of (he pin. 



144 Lessons in Horology. I 

to the movement indicated by the arrow, and the gearing would 
become, on that account, "interior." 

214. Lantern gearing can be interior and then admits of two 
arrangements, according as the interior wheel or pinion carries the 
pins and the other the teeth, or as the large wheel carries the pins 
and the pinion the teeth. 

215. Take, again, as a second example of the application of 
this construction, the straight line A B, given as the form of the 
pinion leaf, and let it be required to determine the curve of the 
wheel tooth (Fig. 53). 

During the movement of the primitive circumference ol the 
pinion around that of the wheel, the line j4 5 will occupy succes- 
sively the positions A' B' , A" B", A'" B'", etc. From the points 
ol tangency, i, 3, 3, 4, etc., let us draw respectively the perpen- 
diculars to these lines and through the points a, 6, c, d, thus 
obtained, let us make a curve pass tangent to the successive 
portions of the line A B ; v& will thus obt^n the form of the 

Let us remark that if the wheel is animated with a movement 
to the right, the position A"" B"" can become impossible for the 
transmission of the movement, for the reason that the wheel could 
then turn without driving the pinion. This shows us, moreover, 
that there exist limits beyond which the driving of the pinion by the 'I 
wheel becomes practically impossible. I 

216. When the line A B passes through the center of the 
pinion, the curve of the tooth is an epicycloid produced by a 
ptnnt of a circle whose radius is equal to half that of the primi- 
tive circle of the pinion. 

217. Note. — On comparing the two methods of determining 
the forms of contact which we have just examined, one can 
prove that the graphical method (209) is necessarily analogous to 
that of the envelopes. In short, in order to obtain the form of the 
tooth, we make one of the primitive circles with the given form 
roll around the other ; the curve sought is, therefore, in both cases, 
that which passes through the meeting point of the normals with j 
the given curve, in each of its successive positions. The reason < 
which has made us separate these two parts of the same whole i 
simply the greater clearness in the explanation of the subject 

218. Let us take, for a last example, gearings formed by the j 
evolvent of a circle. 







• • • • 

L • > I • 

« « 
t « 

« • ! 
» • 

\ \ I : • 

, « ". • • 

V •!* 

•% 4» t 


Vig. 58 


Lessons in Horology. 

The ' ' evolvent " of a curve is another curve, 
C C C" C" . . . produced by a point of a tangent to the first 
curve, whose contact changes continually, in such a manner that 
the distance oi the de- 
scribing point from the 
point of contact may 
be constantly equal to 
the space traversed by 
the point of contact oa 
the curve. Thus (Fig, 
54). -ff' C, B" C", . . . 
being positions of the 
tangent, one should have 
B'C ^ B'C; B"C" = 
B" C, etc The curve C B' B" . . . on which the tangent rolls 
is the "evolute" of C C C" . . . 

The point C where the evolute meets its evolvent is the origin, 
219. For gearing formed by the evolvent of a circle, one adopts 
for the motive tooth the evolvent of any circle concentric and 
interior to the primitive circle of one of the two wheels. The 
profile of the corresponding tooth for the other wheel will be 
determined, then, in a very simple manner. 

Let the evolvent Z? i? of the circle E'E' (Fig. 55) be given. 
In order to determine the point of contact M of this curve and ol 
the form sought, draw the normal A A' from the point of tangency 
of the primitive circumferences ; by the construction this normal is 
at the same time tangent to the "evolute" circled' E' . But if 
from the centers O and O* we draw the perpendiculars O B and 
(y B on this normal, we will obtain two similar triangles whose 
homologous sides are in the same relation. But the sides a O, 
a O' and Cy B' are constant, being the radii of invariable circles ; 
therefore, B must be also constant. Consequentiy, the normal 
A A' of the curve D D sought remains always at the same distance 
from the center O and it, therefore, envelopes a circle E E con- 
centric and interior to the primitive circle of the wheel and whose 
radius is found with that of E' E' in the same relation as those of 
the primitive circles themselves. The form of Ike tooth is, therefore, 
another evolvent of a circle. 

The point of contact being found at any instant on the line 
A A', this line is the geometrical place. 



Fig. 5S 


Lessons in Horology. 

Thus, in gearings of this kind the place of the points of con- 
tact is the common tangent to both circles of consiruction. 

The common normal retains, therefore, a fixed position in 
space during the movement of the two wheels around their respec- 
tive centers. 

This right line can make any angle with the line of centers ; 
it is the general custom, however, to place them at 75° from each ■ 

The especial advantages of this system of gearings are, first, 
that the two wheels being similar and the teeth not showing any 
change of curvature at the passage of the line of centers, any one 
tooth will drive the other before as well as after the line of centers. 
Moreover, the construction of a wheel not depending in any way 
on that which it should drive, all wheels evolvents of circles 
can gear together ; the relation of the velocides which they have 
is only to be considered. This is a valuable property which 
allows a single motive wheel to drive at once several others, or 
to make several wheels gear together successively, as is the 
case in the screw-cutting lathe. Another advantage to be con- 
sidered is that the distance of the centers can vary betvreen cer- 
tain Hmits without the regularity of the gearing suffering in 

The gearing of evolvents can be interior ; the form of the 
teeth, in place of being convex, is then concave. This fact is an 
inconvenience which makes this combination little used. One can, 
in these cases, diminish the concavity by multiplying sufficiently the 
number of teeth. 

220. Third— Roller Method. The principle of this method 
differs from the preceding, but is just as general. 

Let us imagine, first, any polygon, A B C D E FG (Fig, 56), 
tnpelled to roll without sliding the length of a line X Y. At a 
certain moment of the movement 
one of the angles. A, for example, 
is found in contact with the line 
X Y. During the rolling around 
this point all the points of the 
polygon, and with them all those 
which, interiorly or exteriorly, could 
be unalterably connected with them, 
describe arcs of circles around the 

Gearings. 149 

center A. As, for example, the point H, exterior to the polygon 
^tJt unalterably connected with it, will describe an arc a b during 
^e instant of the rolling considered. 

The radii of these diverse arcs of circles will be their normals 
' ^^id will necessarily pass through the point A. 

Let us remark that the length of each arc described depends 
' ^*i that of its radius and on the number of sides that compose the 
F*olygon. If we suppose this geometrical figure formed with a great 
'^Vjraber of sides, the lengths of 
*^-*^e arcs described while it turns 
^-*~otmd one of its sides, will 
*^ iminish. At the limit, that is to 
^^y, when the number of sides 
*^^^^coraes infinite, the polygon is 
^^ unfounded with a continuous 
^^Tirved line, and each of the 
^t^oints which compose it will de- 
^^cribe, nevertheless, as the poly- 
^^on rolls around an instantane- 
^^us point of contact, an infinitely 
^hort arc of a circle. But, how- 

I^ver small it may be, this arc pos- 
sesses, nevertheless, two extreme 
xadii, drawn infinitely near to 
each other and passing through 
the instantaneous center of rota- 
tion. Since they are drawn infi- 
nitely near to each other, either 
of these radii of curvature is, consequently, normal to the point 
considered of the total curve described by this point during the 
continual rolling of the generatrix along the line oi the directrix. 

This established, let there be, moreover (Fig. 57), any curve, 
a b c, which we cause to roll on the exterior of the primitive cir- 
cumference of a wheel and at the interior of that of the pinion. 
If, to be more clear, we suppose that a point H taken outside of 
this curve may be connected with it in an invariable manner, the 
movement of this point will be similar to that of all the points 
composing the given curve. 

During a certain period of the curve's movement at the 
exterior of the primitive circle of the wheel O, this point H will 


Fig. as 

150 Lessons in Horology. 

describe a trajectory d H g ; then, when the movement takes place 
at the interior of the primitive circle of the pinion C, its trajectory 
will be the line /He. These two curves can be adopted as the- 
profile of conjugate teeth. In tact, we imagine that the curve ab c 
follows the movement of the two primitive circumferences in such a. 
manner that these three curves remain constantly tangent at a. 
The trajectories meet at //, since 
this point describes them both ; 
moreover, they are tangent there, 
since the normal for each is ob- 
tained on joining the describing 
point H to the point of contact a 
of the moving curve a b c with 
both of the primitive circumfer- 
ences established. 

Consequently, the common 
normal of the teeth, at their point 
ot contact, passes through the 
point of tangency of the primitive 
circumferences, and the verifica- 
tion of this fact suffices, we know, 
in order to have the curves ob- 
tained, adopted as forms of teeth. 
Let us examine from this 
point of view the following case 
221. Flank Gearings. In or- 
der to obtain a profile very much 
used in the practice oi horology, 
one chooses as the generating 
form the circumference whose 
diameter is the radius of one of the primitive circles and on« 
takes the describing point on its circtimference (Fig. 58). 

In the movement of the generating circle around the primi- 
tive circle ol the wheel, the point A describes an arc of 
"epicycloid" A D. 

In its movement in the interior of the primitive circle of the' 
pinion, this same point A describes a straight line O* A, which is a 
radius of the circle C This plane surface ff Ais called a " fiank. 

Let us remark that the epicycloid which forms the profile of the 
tooth in flank gearings is not the same as that which we have 



detemined for the lantern gearings (21a). In the first case it is 
produced by a point o[ a circle with a radius less than one-half 
that of the primitive circle of the pinion, and in the second this 
curve is produced by a point of the primitive circumference itself. 
222. We are now going to prove that in the rolling of the 
interior of the circle with twice ike radius, the moijing point tra- 
verses a diameter. 

If one represents in effect any position whatever, O", of the 
moving circle during its movement in the interior of the primitive 
circumference of the pinion, the angle inscribed, A' O M, has for its 
measure the half of the relation of the arc A' jT/ comprised between 
its sides to the radius }i A' O, that is to say, ^^■ 

One can, on the other hand, measure it as an angle to the 
center C by the relation of the arc comprised A' A to the radius 
j4' (y ; therefore, ^r^- 

But, if the expression of the theorem is true, that is to say, if 
the point A of the generating circle is carried to M along the 
straight line A C, the two angles A' (7 /1/and A' O A should be 
«qual and superpose ; we would, therefore, have the equality of 
the terms : 

The arc A' M is equal, in fact, to the arc A' A, since the 
rolling of the generating circle is eSected without slipping ; the two 
relations are, therefore, equal and the point M\^ found, in conse- 
quence, on the radius A 0'. 

Since it relates to any instant whatever of the movement, this 
point, therefore, does not leave the diameter O A, which is, then, 
properly the trajectory sought. 

223. If one imagines the flank in any position whatever, as, 
(or instance, O' D (Fig. 59), its point of contact J/ will be obtained 
by erecting to it the perpendicular A M. The angle A M O 
being a right angle, the point M will be found on the circum- 
ference which has A Cy ss diameter ; consequendy, in flank gear- 
ings, the location of the points of contact is the generating ciraiin- 
ference itself. 

224. An analogous reasoning to that which we have de- 
veloped for a preceding case (213), shows that in the simple 
flank gearings the driving can only take place on one side of the 
line of centers. 

152 Lessons in Horology. 

225. In order that the contact o( two similar teeth ma\' 
raence before the line of centers and end on the other side of that liner' ^ 
it suffices ii each tooth has a mixed profile formed with a flank interior" 
to its primitive circle ant^ 
with an epicycloidal part; 
exterior, generated by a 
circle with a diameter 
equal to the radius of the 
primitive circle i 
other wheel. Thus, for 
example (F'g- 60), the 
circle ff' furnishes i 
successive rollings a flank 
a A for the wheel O and 
a curve A D for the 
wheel O. The circle O/' 
furnishes in an analogous 
manner a flank O A ioi 
the wheel O and a curve 
A n for the wheel O. 

This combination : 
called "reciprocal" flank 
gearing. One can, there- 
fore, say that in recipro- 
cal flank gearings , 
driving takes place on 
both sides of the line ^ 

Let us add that the 
form of reciprocal flank 
gearings cannot be em- 
Fig. 00 ployed for interior 


226> Two wheels with plane interior flanks and epicycloidal 
curves exterior to the primitive circles should, according to the 
generation of their profiles, be made especially for each other, since 
a wheel cannot gear regularly in several others of different diame-i 
ters. This inconvenience is avoided for a series of wheels that one 
wishes to make gear with the same wheel, by replacing in the wheds 
of the scries the straight flanks by curves, one chooses for general- 



g circle of these interior curves and of the corresponding exterior 
e of the particular wheel, a constant circle whose diameter dif- 
"s the least possible from the radii of the wheels of the series. 
One encounters in horology an example of this case in the 
gearings of the dial wheels and the setting wheels. The cannon 
pinioQ drives the minute wheel, in which also gears the main setting 
wheel ; this drives, in its turn, the small setting wheel (168). 

An inverse movement is produced when the hands of the watch 
z set to the hour, and it is then the small setting-wheel which 
bives the other wheels. 

One can, in this case, take the circle O" half of the primitive 

de of the cannon pinion, as generating form of the exterior 

jpicydoids of the wheels and aftenvards make this same generating 

I in the interior of each of the primitive circumferences 

uddered, in order to obtain the interior form of the teeth, this 

s then a " hypocycloid " (Fig. 61). 


154 Lessons in Horology. 

In practice, one substitutes very often straight lines for th^e 
hypocycloids, and thus one obtains a general outline recalling that 
of the flank gearings, although incorrect from the point of view ol 
its construction. 

276 a. Determination of tbe Profile of a Tooth Corresponding 

to a Profile Chosen Arhitrarlly (according to Reuleaux l. Suppose 
given abed f/fT ■ ■ ■ i j ^ the profile chosen, A and B the primi- 
tive circumferences of the t«*o wheels whose respective centers are 
Oimd (/ (,K'K' 6i a). 

One draws the normals ag, 6^, f^. . . . jA, ji. ^j, ^k, to the' 
profile given, Through the points a, b. c, d . . . i, J, k. one p; 
arcs of circles described Irom O as center. From ,Sas center 



the lengths Sg, 6^, c^ . . . J, ^k (normals) one describes arcs of 
circles which will determine the intersections VI, V, IV, HI, II, I, 
/i, //i, ///,, IV^. This series of points, connected, form the 
line of the gearing (place of the points of contact). This done, 
from the point O as center, one describes arcs of circles passing 
through the points I, II, III, ... I^. 11^, III^, . . . The lengths 
of arcs S f, r.Z, s.j. 3.4, /.j, taken on A, will be retaken on B 
and will determine the lengths of the corresponding arcs ^i^. 

It ^x- 2i3\. 3\4i. 4ySi (instantaneous centers of rotation). If 
from these last one lays off the lengths of the normals a^ 6.^ ^ a b, 
^1 J, =^ b 5, Ci 4i ^ c 4, etc., and if these are connected by a con- 
tinuous curve, one will have the profile sought, ai,i>^, c, ■ ■ ■ ii,j\ ^i- 

226 b. Gearing's by the Evolvent of a Circle. In extension of 
that which we have said about evolvent of circle gearings (219), 
one can further establish, in a very simple manner, the kind of 
generation of the forms of contact by employing the method of the 
rollers (a jo). 

In eSect, the primitive circumferences of such 3 gearing being 
known and the generatrix being the straight line A A' (Fig. 55) 

■ 56 Lessons in Horology. 

inclined on the line of centers and passing through the point of 

langency of the primitive circles, the rolling of this line around a 
tangent circle E E, interior to the primitive circumference of ihe 
wheel, will cause to be described by a point M of this line an 
evolvent D D. The rolling of the same line around a circle 
tangent, but interior, to the primitive circumference of the pinion, 
will cause to be described by the same point M a second evolvent 
ly D', which is the form of the conjugated tooth. 

U is clear that the rotation of a line tangent to a circle is 
effected in the same manner when, according to the condi- 
tion established, it must be accomplished on the exterior of the 
wheel's primitive circle and on the interior of the pinion's primi- 
tive circle. 

Remark : Let us further state the fact that an evolvent of a 
circle is nothing more than an epicycloid described by a point of a 
generating circle whose radius is infinite. 


227. Up to the present, before approaching the details rela^ve 
to the distribution of the teeth on their wheels, we have been 
occupied solely with the determination of the curves, or prt 
of contact, by which these teeth mutually drive each other, without 
determining the points where they terminate. The time has 
come to pay altcntian to these questions. 

From the geometrical point of view, a single tooth could, 
stricdy speaking, suffice for the transmission of the movement ; 
but, in practice, there would result complications and phj-sical 
impossibilities, independendy of the ofistacles also very serious, 
arising from friction. One furnishes, therefore, the wheels with 
several teeth, and it is, for this reason, necessary to make them 
all identical. 

Each tooth has two profiles. Strictly speaking, the posterior 
face could be left any shape ; such are, for example, the teeth 
called "wolf," in some gearings for stem winders. It happens, 
however, often enough, in mechanics, that sometimes one wheel 
drives another, and sometimes it is driven bv the other ; therefore, 
the movement takes place in both directions. It is best, for this 
reason, to construct the two faces alike. The tooth is then 
"symmetrical" with relation to a radius of the primitive circle, 
which is, in some degree, its " bisectrix." 

Gearings. 157 

The teeth being identical and their number a whole number, 

hey, therefore, divide the primitive circumference into a certain 

lumber of equal parts between them, which is called the "pitch" 

lot the gearing (,184). This pitch is subdivided into three parts, 

I 'Ca^fitll, the blank and the play. The full is the space measured 

n the primitive circle and occupied by the material of the wheel ; 

tlie blank is the surplus, which should remain dear to allow the 

iDtroduction of the conjugate tooth ; the play is an accessory blank, 

which does not seem at first to be necessarj- from the geometrical 

view-point, but which, in reality, is indispensable. 

Numerous causes render the play necessary for the action of 
EKrings : for instance, imperfections in the divisions of the wheels 
by the machine, the shake necessary to the pivots in their holes, the 
expansion of the bodies of which the mobiles are formed and the 
inevitable introduction of foreign bodies in the wheel teeth, are all 
io many reasons for this necessity. 

228. It can be said that the play is the relation of the arc not 
occupied by the sum of the breadths of the tooth and of the leaf, 
to the pitch of the gearing. 

If we represent by p the pitch of the gearing, by a the length 
of the arc occupied by the tooth on the primitive circumference 
3"^ by d the length occupied on this same circumference by the 
^H'Kponding tooth of the other wheel, the play _;' will be expressed 
''y the formula 

If, for e 

niple, we had for a givei 

p = 6 mm., a ^ 2.8, and b = 2.8, 

one would obtain the play of the gearing 

6 — 1.8 — 2.8 , 
J - >- = -h- 

229. To determine the quantity of play necessary for a gear- 
ing, one has to examine hvo conditions : First the solidit>' of the 
wheel teeth and then the space to be reserved for the free passage 
of small foreign bodies, such as dust or the particles which, i 
tably, are detached from the bearing surfaces on account of \ 
One or the other of these conditions can have the predominance, i 
according to the nature of the gearing. 

158 lessons in Horology. 

Thus, in the gearing of stem -winding works, of the selCS 
gear, the rack of a repeating watch, etc., the conditions inhe 
to the sohdity ol the wheel teeth should evidently predominate. 
the gearing of the fourth wheel with the escape pinion, there 11 
necessarily be reserved space for foreign bodies. 

The same two conditions should also guide us in the choice 
the fonn to give to the part of the wheel teeth which forms what 
called the depth of the teeth. Thus, when one desires a solid 
of teeth one chooses in preference the rounded depth, as Fig. 

tf, on the contrary, one wish^ 
to construct a set of teeth lea- 
ing place for foreign bodie 
one will adopt a form such s 
is indicated in Fig. 63 ; in tU 
last case, one can also use th 
lantern gearing. . 

230. In the gearings 
stem -win ding mechanism, 
change wheels for setting, etc. , one divides the play equally betwfl 
the two wheels ; in the gearings of the train of thi 
deducts it from the pinion 
leaf alone, for the reason that 
this last mobile is made of 
tempered steel and offers, 
consequently, more resistance 
than the brass ol which the 
wheel is made. The solidity 
of the leaf is, furthermore, 
also increased by its greater 
transverse length. 

Another reason which makes us deduct the play from the leaj 
is that the wheels generally drive the pinions ; consequently, it i 
the profile of the tooth which drives the flank of the leaf after th 
line of centers. The curve of the tooth, therefore, must be c 
sufficient length to be able to drive the flank far enough to prevent 
as much as possible, the tooth following entering into contact befon 
the line of centers. This, however, is not always possible for thi 
pinions of low numbers. 

231. For the mobiles of the train, the general rule adopted, 
to give in the wheel half of the pitch to the tooth and the 


Gearings. 159 

Jalf to the blank. The pinions of 6, 7, 8, 9 and 10 leaves would 

pen have one-third of the pitch appropriated to the leaf and 

lo-thirds to the blank. In the pinions of 12 leaves and above, 

lone would give two-fifths of the pitch for the breadth of the leaf 

I and three-fifths for the blank. 

Thus the gearings with pinions of 10 leaves and below have 
a piay of 

- i6-i6 . 

- i7 - i7 

c. i. 

and the pinions of 12 leaves and above 

= etc.. - A- 

232. For the gearings of the change wheels, one can admit y'5 
* play. 

233. For those of the stem-winding mechanisms, one can be 
Content with ^ of play. 

Tturd — Total Diameten. 

234. Before entering into the details relating to the determina- 
"On of the total diameters of the mobiles in a gearing, we 
Commence by the geometrical study of the curves employed in 
^Orological gearings. The principal among these we know to be 
*l*e epicycloid. As 2 preface to this question, let us establish, first, 
*He theory of the cycloid. 


235. Defmition. The cycloid is a curve described by a point 
*3f the circumference of a circle which rolls without slipping along 
a straight line. 

This curve is employed in the rack gearings (191), which 
establish a connection between a uniform transfer and a uniform 
rotation around an axis perpendicular to the transfer. This is then 
the particular case of gearings around two parallel axes in which 
one of the primitive circles, having its radius infinite, becomes a 
straight line. 

236. Drawing of the Cycloid. Let it be desired to describe 
by points the cycloid generated by the point A oi a circle with 

—-diameter D (Fig. 64). 

One draws a straight line A A' equal to the base w D oi the I 
One describes the circle O with the diameter D tangent at I 

l6o Lessons in 

the point A to the line A A'. One divides the generating circum — 
ference and the base into the same number of equal parts, 12, foi — 
example, which are numbered in the manner indicated by the figure. 
From the point of the center one draws a straight line parallel tam 
the base ; this line will contain the successive places transveraed by— 
the center of the generating circle during its rolling. Let us indicate? 
on this parallel the positions of the center O, corresponding to the- 
positions of the circle when it is in contact with the base at the^ 
division points, and, from each of these centers, let us describe:- 
circumferences with diameters equal to that of the generating- 
circle. 4 

Let us remark that when the generating circle has arrived at \ 
the center i^, the point ly of its circumference is lowered to /, and ' 
the point A, whose movement describes the cycloid, should be ele- 
vated to a height equal to the distance which the point / , is lowered. 

Thus, in this new position, the point A should be found on 
the circumference whose center is at i ^ and also on a line parallel 
to the base passing through i^. 

In the same manner, we could determine the successive 
positions occupied by the point A while the center of the gene- 

rating circlt is found at 2^,, jo. 4d> ^'^■> ^""^ ^^ connecting all 
the points thus obtained by a continuous line, one obtains the 
cycloid A B A' sought. 

237. Drawing of tbe Cycloid of a Contiauons MoTement. One 

understands that the circle (7 is a circular plate on the circumference 
of which a point or pencil A is fixed (Fig, 64). If one causes the 
plate to turn without slipping along a straight rule whose edge 
coincides with A A', the point or pencil A will describe the cycloid 
of a continuous movement. 


238. Normal and Tangent to the Cycloid. Let M be any 

poim whatever on the cycloid A M B (Fig. 65), through which 

It is desired to draw a normal, then a tangent. Having traced 

the base A A' and its parallel E E' containing the places occupied 

successively by the centera of the generating circle during the 

rolling, we will find 

■lie center O of the 

generating circle 

corresponding t o 

'fie point M of the 

cycloid, by tracing 

'^om the point M 

"'tH an opening of 

•"^ compass equal 

*° tbe radius oi the 

f^n crating circle, an 

^^ of a circle pass- 

'"§■ through the line 

^ ■£'. The point 

intersection O will be the center of the generating circle. 

Dropping from the point O a perpendicular on the base A A', 
"*^ point P will be the momentary center of rotation of the gene- 
"^ting circle. Its movement is composed of a movement of trans- 
action parallel to the base and of a movement of rotation around 
'^ center O. The point P being thus the center of this combined 
'*iovement, the point M will describe an arc of a circle infinitely 
siJiall around this momentary center ; the straight line MP being 
the radius of this arc will consequently be the normal to the point 
^ sought. The tangent being perpendicular to the normal, should 
pass through the point T of the generating circle ; one knows, in 
fact, that every angle inscribed in a semi- circumference is a right 

239. Evolute and Radius of Curvature of the Cycloid. The 
evolute A' J A of the semi-cycloid A B is a. semi-cycloid equal to 
its evolvent (218), 

Let A A' be the base (Fig, 66), B/ the axis of the cycloid 

generated by the point j1/ of the circumference T M P. Let us 

describe a circumference on the diameter/"/"^ T P \ through 

the point P' draw £ /^parallel to A A' ; then draw the lines M M' , 

_ ^ rand M' P\ On account of the equality of the angles MP T 

i62 lessons in Horology. 

and P' P M'. the two right-angled triangles PM 7" and P M' 
are also equal and one has 

M' P^ P M. 
from whence 

M M' = iP M. 

The straight line M M' is the radios of nirvalure of the poin 
M: that is to say, the radius of the circumference which has t«' 


consecutive elements infinitely small, common with the curve at 
this point 

Thus the radius of curvature at any point whatever M of the 
cycloid is twice the portion M P cA the normal comprised between 
the curve and the base. 

Designating the angle M O Phy a, by 8 the radius of curva- 
ture and by r the radius of the generating circle, one has 
8 = 4 r. sin i a 

It is thus easy to see that the foot A' , of the evolute corres* 
ponds with the summit B of the evolvent, while the summit of tho 
evolute blends with the origin of the evolvent. 

240. Length of the Cycloid. The length of the portion A M*- 
of the cycloid A M' A\ is equal to the length of the radius C 
cur^-ature of the point M of the cycloid A M B \ it is equal to 

M M\ since this is the length of the line unwound from the cycloid 

portion A M' . 

One, therefore, has 

A M' = M M' = 1 M P = AT. sin J a. 
In order to obtain the length /' of the cycloid portion A\ M' 
*e have evidently the difEerence : 

/' = 4 ^ - 4 »■ sin ! a = 4 ^ (1 - sin 1 •). 

241. Definition. The epicycloid is a curve described by a 
point of the circumference of a circle rolling without slipping on 
*»e circumference of another circle. 

The generating circle can either roll on the exterior or on the 
*Oterior of the director circle ; in the latter case the interior epicycloid 
'*3 called hypocycloid. 

We have seen that this curve is employed for the form of teeth 
^ the gearing of two wheels turning around two parallel axes. 

242. Drawing of the Epicycloid. This drawing is analogous 
to that ol the cycloid. Let us describe first from the center C the 
director circumference on which the generating circle O should roll. 
Mark on the circumference C a length A A' equal to the length of 
the circumference of the generating circle 0. The latter being 
tangent to the point A, divide its circumference and the base A A' 
Into an equal number of parts, 12, for example. 

From the center C describe afterward a circumference with a 
radius C O ; on this circumference will be the places occupied, 
iSuccessively, by the center of the generating circle ; draw then the 
radii C A, Cj, C,, . . . etc., prolonged to the circumference pass- 
ing through the center of the generating circle. Describe then 
Jrom the points r^, 2^, j„, . . . etc., as centers, circumferences 
with radii equal to the radius of the generating circle. 

Note now that when the center of the generating circle has 
arrived at ig, the point /, of its circumference is lowered to 7 ; this 
point has, therefore, approached C the same distance that the 
point A has been removed from it. On describing, therefore, from 
the center C a circumference passing through the division 7, of the 
generating circle, we obtain the point .4, by the intersection of 
this last circumference with that of the generating circle from 
point /g. 

m the J 


LessoJis in Horology. 

In the same manner we could detennine as many poitii 
wished, and on connecting them by a continuous line we wouL^ 
obtain the epicycloid sought, as it is represented in Fig. 67. 

243. Drawing of the Epicycloid of a Conttniious HoTemei 

C and being circular plates and A a pencil point fixed in 
circumference O, one understands that on making the plate O roU-J 
without slipping on the plate C, the pencil will trace the epicycloid J 
A B A' of a continuous movement. 

244. To Draw a Normal, then a Tangent to the Eplcycioia.J 
Identical considerations to those which have enabled us to draw 4 




tomiiJ and a tangent to the cycloid furnish us the means o[ drawing 
tlese same hnes to the epicycloid. If it is desired to draw a norma! 
■*io the point M of the epicycloid A M B (Fig. 68), we commence 
"fcy seeking the center of the generating cirde belonging to the 
j>oiiit M of the epicy- 
cloid, by laying off ^^ 
irom this point M, the 
Tadius of the gene- 
rating circle, on the 
circumference around / 
which it is moved dur- 1 
ing the rolling of its \ 
center. From the 
point Oj we figure 
the position of the 
generating circle and 
we find the point of 
tangency P, of the 
two circumferences. 
During an infi- 
' nitely small period of 
tiovement all the 
points of the gene- 
ts ting circle, as also 
|those which are un- 
■changeably connected 
v\vith them, describe 
infinitely small arcs of 
circles. Since the nor- Figros 
tnal of an arc of a 
I cirgle unites with the radius, we would have, therefore, the radius 
1 MP as normal to the point M of the epicycloid A MB. 

On drawing afterward from the point jl/a perpendicular to the 
normal, we would obtain the tangent Af T, This tangent should 
pass through the point T of the generating circle, since every angle 
inscribed in a semi -circumference is a right angle. 

1245. Evolute and Radius of Curvature of the Epicycioid. We 
have seen that the evolute of a cycloid is a cycloid equal to its 
evolvent and placed in such a manner that the summit of the 
evolute coincides with the origin of the evolvent and the summit of the 



Lessons i 

evolvent is lound on the same perpendicular to the base as the 
of the evolute. Moreover, the two bases are parallel to each other 
and separated from each other the diameter of the generating cirde. 
As regards the epicycloid, we will see, moreover, that its 
evolute is a similar curve, but not equal. The summit of the 

evolute and the origin of the evolvent at the point A (Fig. 69) 
still coincide ; the summit S of the evolvent and the origin A-^" are 
found on the same radius £ C, but their generating circles are of 
different diameters. 

The two demi -epicycloids are contained in the same angle v4 C B 
which we will designate by t. If we call J? the radius of the base of 
the evolvent and r that of its generating circle, R" the radius of the 

Gearings, 167 

l>ase of the evolute and t* that of its generating circle, we see that 
Ae length of the base of the evolvent is 

^d that of the evolute 

R' % ^mr*. 

On dividing one of these equations by the other, one obtains 

R e _ mjr^ 
or i?^ e ~ w r-' 

R' ~ r^' 
On the other hand, one should still have 

, R — R' = 2r', 

"^^^yci whence one draws 

, R — R' 

r* ■■= 

^^ substituting ^ 

R 2r 

nd R' R — R' 

(I) R' = 

R -\^ 2r 

Let, for example, R = 60 mm. and r = 20 mm. , we would 
^^X/e, in this case, 

60 + 2 X 20 100 ^ 

^nd 60-36 

r' = ^— = 12 mm. 


246. The point M' (Fig. 69) is the center of curvature of the 
I^C)int M of the epicycloid A M B \ it is situated on the evolute 
-^1" M' A, 

We have, in effect, 

arc M P = arc A P, 

^ince the generating circle has rolled without slipping on the cir- 
cumference A A'. But, 

arc M P =- ra and arc A P ^ R X angle A C P 

Let us call the angle A C Py y and place 

ra = Ry, 

one will then have 

_?1 — JL 

1 68 Lessons in Horology, 

When the generating circle with radius t* has rolled without 
slipping on the base B' -^i", this length of arc B' A^" is equal 
to ir r* . One has also 

ir r' = arc /* iJf^ + arc M' P\ 

arc P' M' = arc P' A/', 

arc /* M = arc S^ P', 
and as 

;^xz P M' = r* Y. angle P (y M\ 
arc B' P' = R' 7, 
one will then have 

r* X angle P (y M' = R' 7, 
from whence 

But as we have 

we will also have 

R' angle P (y M' ' 

iL — JL 
R' R' 

•y _ If 

a ~ angle P (y M" 

from whence a = angle P Cy M\ 

The point M' thus determined belongs, therefore, properly to 
the evolute. 

Since the angle M TP = M' P' /* = ^ a, and since the angles 
at Ma.nd at M' are right angles, the straight lines MPsind P M' 
will have the same alignment. 

247. The straight line M M' representing the line developed 
is the radius of curvature of the point M of the evolvent and the 
length of the arc A M' developed. 

We have, in effect, 

M M' = M P -\- P M' \ 


M P ^= 2 r. sin \ a and P 3P = 2 r'. sin ^ a ; 


M 3r :--- 2 (r + r^) sin J a. 

Designating the radius of curvature by 8 and replacing r' by 

R_^__R' _ ^ / _ R \ 
2 2 V^ R -\- 2rJ' 

we will have 

(2) 8 - 2 r I I -f -=5-- ) sin , tt — 4 r -^— 1 sni -?, a. 

For a numerical example, let r ^^ 20 mm., ^ = 60 mm.» 
a 60° ; we will obtain successively. 



Log. 64 = 1. 80618 
+ " sin J a = 9.69897 

Log. G :^ 1,50515, from whence s = 32 mm. 
For J^ a ^^ 90°, the radius oi curvature 8^= -^i". -5^^ 64 mm. 
248. We know that in flank gearings, the radius of the gene- 
rating circle of the epicycloid is equal to half the primitive radius 
of the pinion which we will designate by r' ; one has. therefore, in 
this case (a), ^^ — ^ 

The radius R takes, then, the notation r, primitive radius of the 
wheel. The angle 5^ a is the angle formed by the flank of the 
pinion leaf and the line of centers. Under these conditions, the 
mula (2) becomes 

n and n' being the numbers of teeth (184), 


249. The formulas (2) and (3) show that at the origin, the 
radius of curvature is nothing. This fact indicates that at this 
point the curve is united to the primitive radius of the wheel 
without forming an abrupt angle. 

The radius of curvature increases, afterward, proportionately 
to the sine of the angle formed by the flank of the leaf and the line 
of centers ; it becomes greatest when the angle J^ a is equal to 
90° ; it diminishes then to become again zero Sot j4 a- = o. that 
is to say, at the point of the curve's inflection. 

250. Length of the Epicycloid. The length of arc of a curve 
is equal to the length of the line developed. Thus (Fig. 69) 

'(■ + ^T^)- 


^H One has most frequentiy occasion to determine the length of 
^B an epicycloidal arc calculated from its origin ; it is, therefore, 
^■necessary to determine its length from the point .^j". For this 
^B purpose it is equally proper to take in place of the angle P P' Sf 




Lessons n Horoiogy. 

its cooqilaneiitarf P' P AT, whidi we vill des^nate by | p. For 
the point M" the ai^le ^ f becomes thus equal to P" yt' M" irhicb 
obliges D9 to change the sioe ioio costDe. 

To obtain the length A-^" M" d the eptcydoid W," M" At 
let 03 remulc that this length is equal to A A^" — A M". 


" '="(-^fl7)(--»>0; 

r is here the radius of the generating circle of the epicycloid A MB 
and R the radius of its base ; so we have (245) 

On substituting these two values in the above equation, we 
will obtain the length of the epicydoidal arc, calculated from the 
point of origin, thus : 

U) /=4^(. + ^)(.-«»4p). 

251. Brst AppUcatlon. A wheel of 80 teeth can drive the 
leaf of a pinion with 10 leaves, after the line of centers, an angle 
\% = 34° 45' 48" (^57) ; what is the length of the epicydoidal 
arc of the tooth, starting from its origin when the primitive radius 
of the wheel is 10 mm.? 

Solution : The radius r" of the generating circle of the epicy- 
cloid is T^, X 10. We have, therefore, 

and ^ 

from whence one obtains 



) = 3.5 X 1-0625 — a.6s6a5. 

The natural expression 

of cos ^ p is 

it follows 

Cos i P = 0.831514, 

1 — cos ip - 0.178486. 



■ + ^) = °t=t^' 

f log: 


05 JP ) = 0.25160 — I 

from whence 

log : / = 0.67587 — I, 


= a.4741 mm. 




^^P Remark. — The height of the ogive is equal, in this case, to 

T 0.42285 mm. We indicate further on, the means of calculating 
■"/ "lis latter value (258). 

/ 252. Second Application. A wheel of 60 teeth can drive the 

( 'eai of pinion with 6 leaves, an angle 2 P ^ 4^° 15' 47" after the 
toe of centers ; what is the length of the epicycloidal arc of the 
^_ tooth, the primitive radius of the wheel being 5 mm,? 
^^B Solution : We have here r" ^ o. 25. Therefore, 


i + ^ = 1.05 4 '^ = I 


cos IP = 0.74006 
I — COS i P =0.25994. 

log: 4»' ( I + ^) = 0.021 19 
+ log : (i — cos i p ) = 0.41487 — I 
log : / ^= 0.43606 — 1 
/ — 0.27293 mm. 

Remark. — The height of the ogive is, in this case, equal to 
-*- 2325 mm. 

253. Third Application. Similar problem for the gearing of 
^ wheel of 70 teeth in a pinion with 7 leaves, the angle ^ p being 
^S" 55' 15" and the primitive radius of the wheel 5 mm. 

Solution : We have 

log: 4'^( 1 + ^) = 0.021 ig. cos Jp = o.766g3 

log : { r — cos J p ) ^ 0.3674 9— i i — cos ^ p = o. 23307 
log: / = 0.38868 — I 
and / =; 0.34472 mm. 

Remark. — The height of the ogive is, in this case, 
0.21 165 mm. 

The calculation is, therefore, the same for all the Hank gearings, 

I it is useless to follow further examples of the application, 


254. The radius vector C M (Fig. 70) which we will designate 
by 8, forms with the initial radius vector C A ^ R, radius of the 
base, an angle S ; one can conceive that there should exist 


Lessons in Horology, 

a relation between the radius S and the angle 6. This relation is 
complicated, but it has a great importance in the calculations 

relative to the de- 
. termination of the 
;' total radius of the 
; wheels. 

• If the angles 
Ma A = a and 
M C a = p are 
/ known, we would 

have the propor- 



•••• j-^v-:-^-^^"- 

Fig, 70 

and in the second 

sin a 

r sin P 
from whence 

8 = r 

sin a 

sin p ' 

Let us now 
seek for a relation 
between the angles 
6 and a, and for 
this purpose pro- 
ject the point i^on 
the straight line 
C d^; we will thus 
form two right- 
angled triangles 
In the first we have 

M E = r, sin a, 

from whence 

M E = {R -\- r — r. cos a,) tang. R 
r. sin a ^ (^ + r — r, cos a) tang. p. 

sin a 

On dividing by r, it becomes : 

= {—- + I — cos a^ tang. p. 

A A' = MA\ 
r a = ;? (0 + P): 


P Gearings. ff^ ^ 

M from whence 1 

( P-:^--"' 1 

then 1 

(5) sin a = (^^ + 1 ^ cos « ) tang. (-^ a - 8 ). 1 

255. Remark. — One can also project the point O" (Fig. 70) 

on the prolongation oi C M, and one thus forms the two right- 

angled triangles (7 H M^nA (7 H C. In the first case, we have 

O' H= r. sin (a + P). 

and in the second 

0' // = (^ + r) sin p ; 11 

from whence ■ 1 

r. sin (a + P) = (^ 4- r ) sin p. J 

On replacing n by its value ^^^^1 


and on dividing by r. it becomes ^^^^H 

sin[A(, + s, + p] = (A + ,),i„p, ^1 

which one can also write \ 

(., si„[A,+ (A + ,) p] = (A + .),:„ 5. 

256. The calculation of the equations (5) and (6) is compli- 

cated ; one sees, in fact, that one can only proceed by successive 

approximations. We give below an example of this kind of 


257. Nnmerlcal Application. To find the value of the angle . 

(Fig. 70) corresponding to the posidon of the point M in the epicy- 

cloid of the tooth of a wheel with 60 teeth gearing in a pinion with 

6 leaves. 

We will suppose that the point M considered belongs to the 

point of the tooth ; it is, therefore, the extreme point of the curve 

of this tooth. 

The application of the formula (5) gives us, first, \ 

-ff , 60 »■ 3 

- + ._-_+,_„; _ = J-_„.„s. 



since half of a tooth should take up a quarter of the pitch of the 

^ _ J 

Lessons in Horology. 


us suppose 

first, the angle 


= 80 

; we would then have 

P = 

i- '-'->• 

• so' 

= a°3o'. 

The calculation gives 

V + ^ = 

21 log : 

+ I 

— COSa = 

= 1.31861 

cos 80° = 




tang: p = 

= 8.64009 

+ I 

— COS a = 



The logarithm of the second member of the equation (5) is, 

therefore, 9.95870 ; in order that equality may exist between the 

first and second members, it would be necessary for the above 

logarithm to be equal to that of sine a = sine 80°. We have 

log : second member ^ 9.95870 

log: sin : 80° = 9- 99335 

diflerence = 0.03435 

The equality of the two members of the equation (5) is not 

verified ; the logarithm of sin a. is too great by 0.03465 ; it is 

sequently necessary that the value of a be greater than 80°. Let 

us try to take a •= 86°. We will have 

p = -j^ a - 9 = s° 48'. 

'^- + I = 21 log : -- + I — COS. a = 1.32077 

— COS : 86° ^ 0.06976 log : tan : p = 8.68938 

20.93024 10.01015 

log : sin » ^ log sine 86° 9.99894 

Difference, .01 

This time, the logarithm of the first member is smaller than 

that of the second, which indicates that the angle of 86" is too 

great The angle a should, therefore, be found between 80 and 

86 degrees. 

Let us now establish the following proportion, taking note that 
for 6° of arc the difference between the natural values of the cosines 
is 0.04586 : 

0.04586 6 , . 6 X H2I 

■ — - ^ — from whence :i: ^= ■■ ■ „, — , 
0.01 IS I X 4586 ' 

on making the calculation 

.r = I. =4666 = i" 23'. 

Thus, if the difference which one obtains by the calculation of 
the two members were proportionate to the difference of the angles 






Anglo of Driving 



Angle of DrtTing 


Wheel 60 
Pinion 6 

42= >5' 17" 

17° 44' 'i'f 


Wheel 70 
Pinion 7 

39° 55' 15" 

11° SC/ 37.857* 

51° 25' 42.857* 

Wheel 60 
Pinion 8 

37° 36' 20" 

r 23' 40" 


Wheel 64 
Pinion 8 

37° 42' 30* 

7° 17' 30" 


Wheel 80 
Pinion 8 

38° o'ss" 

6" 59' 5" 


Wheel 75 
Pinion 10 

34° 39' 53" 

jO ^„, ^„ 


Wheel 80 
Pinion 10 

34° 45' 48* 

I- 14' .2" 


Wheel 90 
Pinion 12 

32° 27' 30" 

32° 27' 30* 
in place of the 
30° necessary 

Wheel 96 
Pinion I a 

3^° 33' -4' 

. 32° 33' 14'' 
in place of the 
30° necessary 

Wheel uo 
Pinion IS 

33° 50' 

in place of the 
30" necessary 


Lessons in Horology. 

chosen, which is not entirely the case, the angle of S6° would be 
i" 28' too great, which gives a new value for a, say, 86° — 1° 28'^ 
84° 32'. Let us begin again, therefore, the verification for this 
last value. One has 

a ^= 84° 32' and consequently p = 2° 43' 36" 
|-i — 31 log:— -+i — cosa= 1.32024 

^ 0.09527 

-(- log : tang : p ^ 

log : sin a = 


One sees that we have very nearly approximated the real 
value of a ; if it Is desired, one can approach it still nearer by a 
new approximation and one arrives at length at a = 84" 31' 34" 
and p = 2" 43' 34". 

Calculatian of the Total Rsdhn of the Wheel. 

258. Knowing, now, the two angles a and p, it becomes easy 
to calculate the value of the total radius of the wheel ; that is to 
say, of the radius s C^ig- 1°)> ending at the point of the ogive of 
the tooth. (See preceding calculation : 257. ) 

Making this calculadon for the primitive radius R = 1, 

would have : 

= 0.05 R 

for the data of the preceding calculation. 
One should then have (254) 

^ ^ °"5 St 


log : sin : a — 9.99801 
log : sin : p — 8.67728 
-|- log : 0.05 ^ 0.69897 — 2 

log : S ^^ 0.01970, from whence G = 1.0464. I 

Form of the Exceii of the I^nion Leaf in a Flank Gearing. 

259. The gearings of the wheels in a watch always turn in the 
same direction,* and it is the teeth of the wheels that drive the 
pinion leaves. 

• ET(M!pl. howpter. Oie setting wheels, which ore driven by tUt 
nim. In Ihia mecliaulani. the minute Bheel does not drive the cam 
nioment when this Iraia ii lurnad \ij the band in setting the watch t 

Gearings. 177 

The equation (5) can be written under the following form, 
remarking, however, that the primitive radii are proportionate to 
the number of teeth and that, in this formula, the radius r of the 
generating circle is equal to half of the primitive radius of the pinion. 

sin a = (-^ + I - COS. a ) tang. (-^ « - 6 ). 

When in a given gearing one finds through the calculation of 
the above equation, the ang 



the tooth of the wheel may drive the pinion leaf a sufScient quan- 
tity after the line of center, so that the contact of the following 
tooth may commence on this line. Such is, for example, the case 
of a wheel with 96 teeth gearing in a pinion of 12 leaves. 
We have, in fact, in this case, 

ka = 65° 6' and J a = 32" 33'. 
The tooth can, therefore, drive the leaf an angle of 32° 33', 
On the other hand, we see that the angle which separates two con- 
secutive flanks is 

T? = 3°-- 

Consequendy, one can prove that the tooth drives the leaf 2° 33' 
farther than is absolutely necessary. 

On the contrary, for the gearing of a 60-tooth wheel in a 6-leaf 
pinion, one has • = 84° 31' 34", from whence -^ a ^ 42° 15' 47". 


I -^- 

^ft Since the leaf should be driven by the tooth an angle of 60° 
"and it is driven in reality only an angle of 42° 15' 47" after the line 
of centers, the contact must necessarily commence 17° 44' 13" 
before this hne, since 42° 15' 47" + 17° 44' 13" — 60°, 

The gearing should be arranged, in this case, in such a 
manner that the tooth enters into contact with the leaf before the 
line of centers, and we have seen that, for this purpose, the leaf of 
the pinion must be terminated with an epicycloidal form susceptible 
of being driven by the flank of the tooth up to the moment when 
the contact is made on the line of centers ; starting from this point, 
t is the curve of the tooth which drives the flank of the leaf (225). 

1 78 

Lessons in Horology. 

The epicycloid of the leal should be described by a point oi » 
generating circumference whose diameter is equal to the primitivi 
radius of the wheel. 

260. Since the angle at which the tooth should enter into cow 
tact with the leal before the line of centers is never very consider-, 
able, it is very rarely necessary that the excess of the tooth be per- 
fectly ogival. This shape would be, moreover, rather hurtful thm 
useful, because the friction would be increased and it would also 
necesitate longer teeth for the wheel, in order to allow the Era 
introduction of the pinion leaf in the space which separates twt 
teeth of the wheel. 

One preserves, therefore, only that part of the epicyclcA 
which is directly useful. In large mechanics, one simply remove! 
the desired quantity from the points of the teeth.* 

In horology, one terminates the pinion leaf by a rounde; 
form, an arc of an ellipse, for example. This shape of the excest 
should be determined in such a manner that its radius of curva- 
ture at the point of connection with the curve of the epicycloid, 
should be the same for the two lines (Fig. 71). It is evideB 
that, for security, one makes the contact oi the tooth and the lea 
commence some degrees sooner than is necessary. Thus, in thi 
preceding case, one will admit a contact commencing 20° before th^ 
line of centers, rather than 17° 44' 13". 

26l> Practically, in a great number of ordinary pinions, o 
finds the excess of the leaf terminated by a half circle. We 2 
really in a position to recognize that this form is defective, 1 
cially lor pinions of low numbers. It is easy to prove that widi 
this shape, the tooth drives the leaf too far after the line of centers, 
that the point of contact does not rest on the circumference of th« 
generating circle and that there can be established, before the line 
of centers, a contact of the flank of a tooth with the rounded part 
of the leaf. There results from this last act a butting often m 

It is probably this butting observed by horologists, which 
frequendy makes them exaggerate the importance of the friction of 
the gearings before the line of centers and which has made them 
believe that this last is much more considerable than that which | 
is produced after this line. The friction observed should only be! 
in reality the butdng, and it suffices to suppress this to per- J 

•Tbli openUou U called tbo chamferiDg of tbo (eelb. ^k 

0' '79 

cepiibly diminish the difficulty 
oi the driving. The calculation 
proves, in fact, that the differ- 
ence between what is called the 
"entering" friction and the 
"leaving" friction is not great 
enough to be so easily found. 

262. In order to determine, 
geometrically, the form of the 
excess of a pinion leaf, it is, 
thereiore, established that this 
form should be composed of an 
epieydoidal arc sufiiciendy long 
'o be able to enter into contact 
with the flank of a tooth of the 
*heel several degrees before the 
point of the succeeding tooth 
leaves the circumference of the 
generating circle. Two teeth can, 
therefore, drive at once two leaves 
through a small arc. 

The symmetrical epicycloids 
of the leaf are afterward termi- 
nated by an elliptical shape con- 
necting them. Suppose, then, it 

\\% i 


be desired to determine this curve, 


which has no other condition to 

fulfill except that its radius of 

Wi ^ 

curvature at the connecdng point 
must be the game as that of the 

' m 

same point of the epicycloid. 

263. Let us determine, first. 

\\ ■ 

the radius of curvature of the 

epicycloid at a point correspond- 
ing to a driving of 20° before the 
line of centers for a wheel with 

\l 1 

60 teeth gearing in a pinion with 
6 leaves. 

V 1 

Since the generating form 
of the epicycloid of the leaf is 

I0 — ^ 

I So Lessons in Horology. 

a circle with diameter equal to tbe primitive radius of Ui« wbed, ' 
the formula (248) 

should be written here 

Admitting »^ 

r = 60, r' = 6 consequently 1 • = 60 = »°. 

we would then have 

6 = 60 ( 1 + -^) sin a" = 65.454 X sin a" 

and, making the calculation 

8 = 2.28433. 

264. Let us now seek the value of the angle w, formed by the 
radius of curvature and the straight line passing through the middle 
of the leaf (Fig.71). 

The angle a c (? is a right angle and we have, therefore, 

I £ = a" ; 

The angle a O £, oz 
it follows that 

We have, moreover, the angle 


since in this case the width of one leaf is equal to one-third of the 
pitch of the gearing (231). 
We will have at length 

w — cds^Cyda — 180= — (Tac — aCyd 

265. We have afterward to determine the straight line C/ e, 

joining the center of the pinion to the first point of contact. 
In the triangle G c O' we know the sides 

and the angle 

C 0'= ir+r' 

Gearings. i8i 

We will have, consequently, the value of the angle a O c ex- 
pressed by J ^ sin ^o 

tang: a(yc= ' ^ 

i r + r^ — i r cos 4* 
or 1 I 2 t 

sin 4° 

tang : a (y c = . r^ o 

^ + I -T— — cos 4®. 

The calculation gives 

a (7/ ^ = 190 o' 46/^ 

The side C c oi the triangle O c G will be at last given us by the 

Sin a (y c 
from whence, after making the calculation, 

(y c = 6.4236. 

266. Let us further project the point c on e, on the straight 
line (y Sf designating by y the right line c e and let us determine 
this line. We will have 

y ^=^ (y c ^\n c (y d, 

^ (T £/ = a (T ^r — a (T' ^ = i9*» c/ 46''' — io*» = 9® o' 46^^; 

therefore, ^ _ ^^^^^ ^ ^.^ ^o ^ ^g,, 


J/ ^ 1.00626. 

267. We now know the radius of curvature 8 at the point c 
of the epicycloid, the inclination of this radius to the line passing 
through the middle of the leaf, therefore the angle w and, finally, 
the ordinate y corresponding to this same data. There only re- 
mains, now, to determine an elliptical curve capable of satisfying 
these conditions. 

268. Radius of Curvature of an Ellipse. The equation of this 

curve being 

one obtains 

dy ^ _ b* X 

, d X a* y 


d*y ^ _ b"- 
dx* a* y* ' 

The general equation of the radius being 

8 = 


d* y 
d X* 

1 82 

Lessons in Horology. 

we will obtain successively 

^* X* \* 

• ==* A« 

2LJ— ^* 


a« J/' 



We have the length of the normal M N (Fig. 72) expres 


Flff. 79 


M N = 

* X* 

a" y 

iif ^ 

One can, therefore, place 

8 = 

a^ M N^ 

we have, on the other hand, 


S = 

sm a/ 


^* sin' w ' 

Gearings. 183 

fron:x whence 


In order to determine the value of b, let us remark that 

p N= — ^ — 
tang w * 

P -A^ being the sub-normal, the general equation of which is 

e d y b* X b* X 

-^ d X -^ (^ y a* * 

froxn whence 

y __ b* X 

tang.ze; a* 

a* y 
b* tang w ' 

Substituting the values of x and of a' in the general equation 

a*y -f b^ x*=^ a*b\ 

^^^^ch we can write under the form 

^'V^ewill have 

X* g* y* S sin* w S sin w cos* w 

«^ ^ b^ tang * w ~^ y tang * w y ' 

*^^m whence 

-___.,/ S sin w, cos * 2g/ \ 


(2) b=^j ^ y 

\ y — hsmw, cos * w 
• The equation (i) gives, moreover, 

(3) a = «• ^5iJ!ali. 

These two last values are those of the semi-axes of the ellipse, 
which fulfills the conditions sought 

269. Numerical Application to fhe Preceding Example. We 

will have 

y = 1.00626. w — 78®. 8 = 2.28433 ; 

* = -%/ ^^ 1.00626' ^ j^^ 

\ 1.00626 — 2.28433 sin 78® cos* 78® 
and , 

a= 1.05834 • JI^^^^ = 1.62245. 

i84 Lessons in Horology. 

270. Total Radius of Ibe Pinion. The total length of t 
pinion leai is composed of the sum of the two lengths C e a 

'P (Fig. 71 )■ 

The distance e p is equal to a — x (Fig. 72). We have 




■05834" X tang 7 
a — jc = 1.62245 — 0.50266 = 1.11979- 
On the other hand, we have (Fig. 71) 

we know (265) 
and the angle (a66) 

O' . 

= a c 

r C e 

O' c = 6.4236 
e O' « = 9° o' 46"; 
6.4236. cos g" t/ 46"= 6.3444. 
The total radius of the pinion will be 

C e = 6.3444 

+ (a-x)^ 1.1198 

R' = 7.4642. 

2?1. Note. — If the excess of the above pinion leaf we 
by a semi -circumference, we would obtain its total radius ii 
more simple manner. 

The primitive radius being 6, the length of the circumference 

is as follows :..,.., ., , 

primitive arcumlerence = 2 « X 6. 

The pitch of the gearing should be equal in length to 2 « 
the number of leaves is 6. The width of a leaf, reckoned on t 
circumference, is one-third of the pitch, therefore 

width of : 
finally, the height of the excess 

leaf = =- ; 

s half of this last value, therefoi 

We will have, consequently, the total radius, 

R'=r' + -^ = S+ 1.0472 = 7.0472 
or ^ 

in round numbers. ~ 


■ Graphical Comtruction of Gearing*. 

I 272. Let us suppose that we know the distance between the 

I centers D, also the numbers of teeth of the two mobiles, and let 

these be, for example, D = 240 mm. ; w = 70 teeth and «' = 7 


The formulas (.185) 

give us the value ol the primitive radii of the 

a(id r" — 34a - 

70 + 7 ■>■ 

the calculations made, one obtains 

r = ai8.i3 .... and ^ = 21.818 .... 
From the centers C and C (Plate I), previously determined, 
"■^e describes the two primitive circles calculated, tangent at the 
P'=»int a 

Let us suppose that we wish to construct a flank gearing, one 
'^n commence by determining the shape of the tooth. Describe, 
^f this purpose, the generating cii'cle O" with radius equal to half 
*^ac of the primitive circle of the pinion, and by its rolling around 
*>e primitive circle of the wheel, let us make it describe the 
^Iiicycloid A B. 

Let us remark that for the clearness of the plan, instead of 
Putting the origin of the curve on the line of centers, as would be 
the logical way to do, we have carried this construction back to 
another point, which, practically, amounts to the same thing. 

In order to construct the epicycloid, one can follow the method 
We have indicated (242), or proceed more simply by carrying back 
on several successive positions of the generating circle, such as 
I, 2, 3, ... , certain lengths i, I = i,o, 2, II = oi, + i, 2;, 
3, III ^ 01, + ij 2, + 2, 3,, etc. On connecting the points 
I, II, III, .... by a continuous curve, one obtains the epicycloid 
of the tooth. 

This method, which is not exactly correct, since it substitutes 
the lengths of chords for the lengths of arcs, is, however, admissible 
for drawings in which the successive positions of the generating 
circle are relatively close together. 

We divide, afterwards, one of the primitive circumferences, 
that of the pinion, for example, into as many parts as that wheel 


lessons in Horology. 

should have teeth. We can commence this division at any point w\ 
the circumference, but generally it is commenced on the line of 
centers, or at the point where the leaf of the pinion should be found 
at the moment of the last contact with the tooth. 

In order to determine this last position, one can make use of 
the table (257), which gives us, (or the gearings most generally 
used in horology, the angle of driving of the leaf by the tooth 
after the line of centers. Thus, for the gearings of a wheel of 70 
teeth with a pinion of 7 leaves, we see that the tooth drives the 
leaf 39° 55' 15" after the line of centers. 

If one wishes to determine this position, graphically correct, 
a previous division of the primitive circle of the pinion gives 
the "pitch of the gearing." One lays oS this length from the 
origin of the cun'e A, to C ; one divides the pitch into two 
equal parts, one of which should be occupied by the whole of the 
tooth, the other by the space (231). The extremity D of the 
Ogive of the tooth will be determined' by drawing the prolonged 
radius O D passing through the middle of the tooth. From the 
center O, one describes an arc of a circle passing through the 
point D and one thus obtains the point d, extreme position of the 
contact of the tooth with the leaf (223). One draws, afterwards, 
the straight line (7 d prolonged to the primitive circumference of 
the pinion. The point i is then the point of departure for the 
definite division of the pinion. If one has proceeded with exacti- 
tude, the angle a O' i should be, in this case, equal to 39° 55' r 
One can thus prove that the first contact of the tooth with the lea!' I 
should take place before the line of centers and at an angular dis- \ 
lance from this hne of at least 11° 30' 27". 

The excess of the leaf should be formed with an epicycloid. 
arc, as we have already indicated {^^g). This epicycloid is th^J 
which is described by a point of a circumference with radius equal J 
to half the primitive radius of the wheel. 

The curves of the teeth being thus formed and their posidons.l 
determined, it becomes easy to construct the gearing, by remarkin 
that for the gearings of watch trains, the leaf of the pinion occupies 
the third of the pitch for pinions of 10 leaves and less, and t 
fifths for those of 12 leaves and more (231). 

One then limits the epicycloid of the leaf, according to what wsJ 
liuvc said, by conserving to it only a length sufficient for the drivin 
til conmience a tittle before the point where the first theoretical 




l88 lessons in Horology. 

coatact should be effected, normally. During a very short instant, 
two consecutive teeth are thus simultaneously in contact, and this 
fact suffices to insure the correct action of the gearing. 

The leaf is afterwards terminated by an arc of an ellipse whose. 
radius of curvature at the junction of the two curves is the same aa 
that of the epicycloid determined. 

One then limits the length oi the flanks of the leaves and teeth 
by arcs of circles with radius sufficient to allow not only the free 
introduction of the teeth and the leaves in the corresponding spaces, 
but also reserving the place which foreign bodies would occupy, 
dust and other matters which are invariably introduced, with time, 
into the sets of teeth. 

The gearing is thus constructed and having made the drawing 
on a suflidentiy enlarged scale, one could deduce from it all the rela- 
tive dimensions for its practical construction, as we will see later on. 

2?3.* Plate II represents the same drawing to a still more 
greatly enlarged scale; the distance of the centers is 3200 mm., 
the primitive radius of the wheel 2 meters and that of the pinion 
200 mm, This design allows us to show more clearly the manner 
in which the contact of the tooth with the leaf is effected before the 
line of centers; the shape of the leaf, represented in dotted lines, 
is semi-circular ; one sees thus that in this case the normal at the 
point of contact does not pass through the point of tangency of the 
primitive circumferences, as is the case for the semi -elliptical shape, 
and consequently the force transmitted has not the value whici 
we determined (195 and the following) 



and that there should be produced a ' ' butting. ' ' 

274. The drawing of gearings of ratchet wheels, setting wheds' 
and dial wheels, etc. , is executed in an analogous manner 
examine later on the several modifications admitted for such wheel 
teeth. In this construction there must also be taken into considera- 
tion the manner in which the " play " is distributed (232, 233). 
Pnctic&l Application of the Theoiy of Gearingi. 
275, In practice there are presented problems of different 
natures in which it is desired to determine the relative dimen- 
sions of wheels and pinions. 

■A« Plnlp II has, for lark of sniM. benn re{iuMi3 oiip-hnlf, Ibp rtlstanee of Ihe cei 
IIW mm.; lh« primitive radius of llie wheel. I m., end Uiat of tlie piuiou 100 mm. 


:it ■ 


Lessons in Horology. 

It is evidenl that, at first sight, the use of a suitable instni-^ 
ment to establish these sizes becomes very important to the work-- 
man, for the reason that it saves him all calculation. We will cites 
the one which is the most exact and at the same time the mostf 
simple to use. 

-^^~> y:^l i; g;^— -t jft 276. The Proportional ■ 

Compass and Its Use. The pro- 1 
portional compass, in its most I 
rational arrangement, is formed I 
of two rule plates, straight, and -I 
divided into equal parts ; they are J 
joined together at one ol thei 
tremities by means of a hinge i 
(Fig- 73)- 

These rules can be fastened iaj 
any position by means of a damping'l 
screw V. 

Their divisions should be ex- 
actly corresponding, equal to each 
other and numbered. The point 
; found at the hinge, summit 
of the angle b O b'. 

The proportional compass is 
based on the fundamental principle 
of similar triangles, in which the 
homologous sides are proportional. 
Thus, imagining the primitive 
diameters of a wheel placed at the 
division of the compass correspond- 
ing to its number of teeth and the primitive diameter of the pinion 
at the division corresponding to its number of leaves, one should 
have the proportion 

b b' o b 

rig. 73 

Since, in a gearing the number of teeth of the mobiles should 
be to each other as their radii, or their primitive diameters, one 
understands that to determine the primitive diameter of a pinion, 
knowing that of the wheel, it suffices to place the latter at the 
division corresponding to its number of teeth and, for this purpose, 
to open the two arms of the compass the proper distance. The 

Gearings, 191 

primitive diameter ol the pinioD should then ccMncide with the 
(^vision which corre^xxids to its number of lea\'es. The proportion 

^ then found to be verified. 

But, as has been shown before, we run against the difficulty of 
'lot being aUe conveniendy to fit the primitive diameters of the two 
Jiiobiles in the ccmipass, since these diameters are only thecH^caL 

The difficulty has been overcome in the following way : 

277. On dividing the primitive diameter of any wheel by the 
dumber of its teeth, we obtain a length which we call ^^dietmetrical 
Pitch^^ of the gearii^. The proportional compass always gives 
^c diametrical pitch by its di\asion 

j^ ^hen the wheel is placed so that ..•-*'* *'• ... 

^^ primitive diameter corre^x>nds in 
^*^e instrument to the divisicm of the 
dumber of its teeth. • V^ 

But, if we measure the height Vji -J... - ?V..g. .. 
^f the ogive a b (Fig. 74) and, on \J^ 
Account of the one which is oppo- 
site, we douUe this value, if we after- 

^2xA divide this figure by the dia- '"^ -^... .. •••' 

metrical pitch, we obtain a quotient Pi^. 74 

which, added to the number of teeth, 

will give the total diameter of the wheel in units of diametrical pitch. 

This diameter is then 

2 a b 

d being the pitch considered. 

On now placing the total diameter of the wheel at the diN-ision 

. 1 a b 

its primitive radius will be by this fact placed at the division n. 
The same for the pinion. 

278. Example. Let us consider a wheel with 60 teeth gearing 
in a pinion with 6 leaves, and let us represent graphically this 
wheel with a primitive radius of 540 mm. 

The diametrical pitch should be 


192 Lessons in Horology. 

Let us describe the epicycloid of the tooth by making a gene- 
rating circle with radius equal to haJf the primitive radius of the 
pinion, roll around the primitive circle of the wheel ; r" being the 
radius of the pinion. One will have 

1'" -- 

So x'a 

Let us now calculate the length of the chord c, which subtends 
the half of the arc occupied by one tooth. We have the formula 


One will have, therefore, 

(^ = 3 X 540 X sin 0° 45' 

log r 1080 = 30334238 
log sin 0° 45' = 8.1169262 
log: (1080 sin 0° 45') = 1. 1503500 
Consequently, one will have 

Chord of one-quarier of the pitch = 14.1367 

Let us lay off this length of / on a (Fig. 75) and draw the radius 

a a prolonged to the point i belonging to the epicycloid of the tooth; 

o 6 is then the total radius of the wheel and a b the height of the 

ogive of the tooth. 

On measuring a 6, we will find it equal to 25 mm. and we will 
have the total radius of the wheel expressed in units of diametrical 
pitch, by the s 


60 + 


60 + 2.77 = 62.77. 

One will place, therefore, the total diameter of the wheel at the 
division 62.77 of ^he compass, so that its primitive radius corres-' 
ponds to the division 60. 

2T9. One could proceed in an analogous manner for the pinion. 
Let us remark, however, that while the height of the ogive of the 
wheel is fi.ted, since it is formed by an epicycloid described by 
a point of a generating circumference with given radius, the excess 
of the pinion leaf is not so easily determined. 

• Ona conid have obtBlned 1hl9 result without ths aid at UisiaotaeXiy, hj noUnf 
the ircf and chords nr emal) snglu dlBer very litUB from eiuh other. 


The form of the excess which one finds in a very great num- 
ber of pinions is that of a semi -circumference with radius equal to 
half the breadth of a leaf measured on the primitive circumference. 
This form, although we know it to 
he t>ad, especially for pinions of low 
numbers, ofTers, however, a ready 
"leans for the calculation. / \ 

Suppose n' to be the number of 
leaves in the pinion. 

The primitive diameter expressed 
'" function of the diametrical pitch 
™1 be likewise w', since it is divided ,- 
"ito as many equal parts as the pinion 
liis leaves. 

The primitive circumference is, 

circumference ^ ir r/ 
^d the pitch of the gearing 
I If this pitch comprises a third 
Tor the full tooth and two-thirds for 
the space, the length of the arc cor- 
responding to the thickness of one 
leaf being double the radius E of the 
^ftdrcle of the excess, one will have fis.'ts 

■ '-^T 

^B There must, therefore, be added a value equal to s to the two 
^^wctremities of the primitive diameter of the pinion expressed in 
^Mlnits of diametrical pitch : 
^H Total radius =^«' -\- — = «' -|- 1.05. 


^B^ Thus, for the pinions whose full pari of the pitch is equal to 
half of the space and whose excess has the form of a semi-circle, 
the total diameter should be stopped at the division corresponding 
to the number of leaves increased by 1.05. 

kFor the gearing which we will consider, of a wheel with 60 teeth 
d a pinion of 6 leaves, one should place the wheel at the division 
.77 and the pinion at the division 7.05. 

H)4 Lessofis in Horology. 

280. Let us again make the calculation for a pinion in which 
the leaf is two-fifths of the pitch (12 leaves and above). 

As in the preceding case, the pitch of the gearing is equal to « 
and the radius of the ogive , _ * . 

" " "s" ■ 

as 3 8 must be added to the primitive diameter, we will have 

' ' - f - "=■ 

Tlic total diameter expressed in units of diametrical pitch is, therefore. 

281. In the case of pinions with the excess semi- elliptical, the 
height of this excess becomes superior to those with which the pre- 
ceding calculations have furnished us. 

Wc have seen that the calculation of this value is complicated 
(859 and the succeeding) ; therefore, without entering into other 
details, we refer, (or these values, to the table which we give further 
on (283, third column). 

Thus, taking up again our gearing of a 60-tooth wheel and 
6-leaf pinion, we find in this table, that the total diameter of the 
pinion expressed in imits of diametrical pitch is 7.4648 ; that is to 
say, 7.5 in round numbers. 

282. Practically, to employ in a proper manner the proportional 
compass, one must, therefore, commence by examining the excess of 
the pinion leaves, estimating it with relation to the breadth of the leaf. 

It one judges, for example, that it is equal to half the thick- 
ness of the leaf, one will add a unit to the number of leaves n' ; if 
the excess appears to be three-quarters of the thickness, one will 
add 1.5 and, finally, if the height is judged equal to twice the 
thickness of the leaf, one will add 2. 

A compound microscope, the eyeglass provided with spider 
lines and mounted on its lower side on a carriage furnished with a 
micrometer screw, allowing the object observed to move in the field 
of the instrument, can measure with great precision the height of 
the ogives of the wheels or of the excess of pinions. In default <rf 
this instrument, the method which we have just indicated is exact 
enough to be used. 

283. The table which we give hereafter indicates the number 
of the division on the compass for the gearings most used in 
horologj-. Thus, on placing an So-tooth wheel at the divisii 















Wheel. . 
Pinion . . 



1. 019676 

1. 14 


1. 104 

Wheel. . 
Pinion . . 







1. 105 

Wheel. . 
Pinion . . 



1. 14 


1. 104 

Wheel. . 
Pinion . . 





1. 105 

Wheel. . 
Pinion . . 




1. 18 



Wheel. . 
Pinion . . 



1. 04016 



1. 104 

Wheel. . 
Pinion . . 





1. 105 

Wheel. . 
Pinion . . 







Wheel. . 
Pinion . . 


9 5 



I- 13 

Wheel. . 
Pinion . . 




1. 2441 



Wheel. . 
Pinion . , 





I. 1995 
1. 139 



1.15 real diameter 

1.1 on pressing 2 

leaves on one side 

and 1 on the other 

Wheel. . 
Pinion . . 






1. 175 

Wheel. . 
Pinion . . 






1. 175 

Wheel. . 
Pinion . . 








Wheel. . 
Pinion . . 




Wheel. . 
Pinion . , 



ig(5 /^isons in Horology. 

83.38 of the compass, a loleaf pinion in which it should gear 
should correspond to the division 11.5 if the excess is of semi- 
elhptical shape, or at the division 11.05 'f ^'s form is semi -circular. 

For a pinion with the uneven number of 7 leaves, one will 
find two indications in the table, one giving the real diameter, the 
Other permitting the placing of the pinion with one leaf pressing 
against an arm of the compass, and the two leaves opposite against 
the other arm. This last measure comprises, therefore, in units of 
diametrical pitch, a total radius increased by the versed sine O B 
(Plate II). 

Alter what we have said, it will be easy to obtain in a graphical 
manner the figures corresponding to gearings not appearing : 
the table. 

284. Verification of a Proportional Compass. The two divided 
scales should be perfectly straight and consequently in exact juxta- 
position when the instrument is closed ; this, one verifies by hold- 
ing the instrument to the height of the eyes and seeing if the twa 
scales are perfectly fitted against each other. 

The divisions should be regular and the zero point should be 
found in the center of the hinge. 

It is also easy to verify this condition with exactitude by taking 
off, with a pair of sharp-pointed dividers, a certain number of 
divisions, 10, for example : on moving, then, these dividers over 
the whole of the part divided, it is easy to assure oneself of the exact- 
ness of tliis condition. Finally, on placing one of the points of the 
dividers on the division 10, one should be able to place the other 
on the center of the hinge. 

This hinge should be made in such a manner that the arms 
can be spread without any jerk, that is to say, with even friction ; in 
no case could any play or shake whatever be allowed at this hinge. 

A compass being thus verified, it could be used with the aid of . 
the given table. 

There exist other systems of proportional compasses, most < 
which dispense with the use oi an accessory table. Let us remark, ' 
however, that the one which we have just described has its principle 
foimded on an exact and rational basis and that the table which it 
requires complicates its use \ery little, if at all 

285. iKlerminatlon of the Distance Between the Centers of a 
Gcarinfr by Means of the Proporlional Compass and of a Depthing 
Tool- Having fastened the proportional compass in such a manner 

lof _ 

irt. ■ 


197 I 



that the primitive radius of the wheel corresponds to the figure 
for its number of teeth, one measures, in this same opening, the 
diameter of one of the arms of a deptbing tool. Let d be the 
division corresponding to this last 
measure. One opens, then, the 
depthing too! until the two arms a 
and b, drawn in section (Fig. 76), 
correspond to the division 

■^ + i. 

This opening then gives the distance 
between centers. 

Elxample ; Having regulated the 
opening in the proportional compass 
50 that the total diameter of a 60- 
tooth wheel is fitted to the division 
62.78 (seethe table), one measures 
the arm of a depthing tool and finds 
that its diameter corresponds to the 
division 8 ; we will thus have 

^^+>- 4., 

the pinion having 6 leaves. The 
opening of the depthing tool should 
then be regulated in such a manner 
that the two arms a and b corres- 
pond to the division 41. 

286. The Proportional Compass and Stem-winding Gearings. 

First — Gearing of the crown wheel in the ratchet wheel : The teeth 
of this gearing should be solid ; this is the reason why only one- 
twentieth of play is given them (223). For the same purpose the 
bottoms of the teeth are made with a rounded shape and the ogives 
of the teeth are shortened. These gearings are epicycloidal ; the 
profiles are formed by epicycloids described by a point of a gene- 
rating circumference smaller than half of the primitive circumfer- 
ences. As we have just said, one does not use the whole of the 
epicycloidal arc for the tooth ; it is sufficient that the contact be 
established three-fifths of the pitch before the line of centers, in 
order to be continued until tluee-fifths of the pitch beyond that line. 

19^ Lettmu im J/aralogy. 

The "flank" of the tooth b no longer a stra^ht line, but a 
l^potycloid described by a point of the same generating drcle 
rolling on the interior of the primitive drcumlerence of the wheel 

Thus (Fig. 77) / fl is the useful epicydoidal arc, while a A is 
any curve whatever shortening the tooth ; in this manner, the height 
of the ogive is not determined ; / </ is a hypocydoidal arc gen- 
erally approaching, very nearly, a straight line. 

To determine the height of the shortened ogive in units of 
diametrical pitch, it is necessary to proceed graphically or by simply 
estimating it by the eye 

Generally, for this gearing, the height of the two ogives placed 
opposite to each other can be taken as 23^ diametrical pitch. 

If, then, n and n' are the numbers of teeth, the crown ^^-heel 
should be fitted to the division 

»' + aX 
and in the same manner the ratchet to the division 
« + aX- 





Let us note, however, that since the crown wheel always drives 
the ratchet, it is preferable to make the first proportionally greater 
than the second ; for example. 

Crown wheel, division . , . . n' -\- 2^ 
Ratchet wheel, division . . . w -j- 2]^ 

287. Gearing of tlie Winding Pinion in the Crown Teetli of the 
Contrate Wheel. In these gearings the axes of the two mobiles 
form a right angle between them. Logically, such a gearing 
should be a conical gearing (311) ; in the practice of horology 
it is sufficient, however, to skillfully simulate it. 

One finds two general 
arrangements of this system. | 

In the first (Fig. 78). »blJ -" 

a ^ is the exterior diameter 

of the crown teeth in the 

contrate wheel ; this is, at 

the same time, its primitive 

diameter, for the ogive of 

the tooth is not to be added 

to the extremity of the radiufe, since the teeth are perpendicular to 

the plane of the wheel. The a:own wheel must, therefore, be fitted 

to the division w' of its teeth and the total diameter c d oi the 

pinion, perpendicular at i on n i to the division n + 2}^, as in the 

preceding case and also for the same reason. Therefore, 

Winding pinion, division n -\- 2}( 

Crown teeth of the contrate wheel, division . . n' 

The second arrangement is found in s 

Winding pinion, division . 
Crown wheel, division . . 

ne winding mechanism. 
It admits of a teeth 
range with the crown 
teeth outside of its 
primitive radius (Fig. 
79) ; in this case there 
must be added to each 
of the two mobiles 
the height of the two 
ogives. One will then 

71' + 2 

2O0 Lessons in Horology. 

288. Gearlne of tbc Sliding Pinion and of tbe Small Setting 

Wbeel. Although one could not make use of the proportional 
compass for the study of the relative dimensions to be given to the 
mobiles of this gearing, and as this determination should be 
entirely a matter of calculation, we give here, however, the theory, 
which will not be found out of place. 

Suppose (Fig. 80) r to be the primitive radius of the small 
setting wheel, R its total radius and w the number of its teeth 


r' the radius of the sliding pinion abutting on the under side of the 
small winding wheel, R' its total radius and n' the number of 
its teeth. 

In the generality of cases, one can admit that 

consequently, ~ ' "' 

t' = R' ~ 0.2. 

If the penetration of the two mobiles is greater, the gearing- 
does not work well. Generally, it is desired in practice to deter- 
mine the number of teeth n' in the sliding pinion. 
The proportion ^ „ 

V ~ "^ 
gives us the value 

By an analogous reasoning to that of (286) one can place 
from whence one draws 


Gearings. 2> 

Replacing r and / by their value in the equation (i). 



I { Ji' ~~ o.a) 

^ IJf' - 0.2) {n + a) 

Example : Let ^ ^: 2.4, jR' ^-- 2 and m =: 18, the formula (2) 

289. Gearings of the Dial Wheels. The sma// and the larg^e 

setting wheels are placed in the proportional compass at the division 
corresponding to their number of teeth increased by 2. Therefore, 
Small setting wheel, division . . . . n +2 
Large " " " ....«' -f 2. 

For the gearing of the cannon pinion and the minute wheel, 
s generally the same ; however, notice must always be taken oi 
I tiie form which the excess has, in the leaves of the cannon pinion. 
If the leaves are terminated by a semi-circular form, it would then 
be necessary, in this case, to place the pinion at the division corres- 
ponding to the number of leaves increased by i. Sometimes, also, 
the teeth of the minute wheel are formed in such a manner that one 
s obliged to add 2.5 or even j to their number, in order to obtain 
I the division of the compass at which this wheel should be placed. 
In order to verify at one time the series of relations between 
the wheels for setting the watch, the cannon pinion must be fitted 
to the division corresponding to its number of leaves increased by 
3, and the other wheels, dial wheels and large and small setting 

I wheels, should likewise correspond to the divisions for their respec- 
tive number of teeth increased by 2. 

If there should be a fault, it is always better that the wheel 
which drives be a trifle large. Since one prefers, in this train, an 
easy and smooth transmission, and since the small setting wheel is 
the wheel which drives, one can allow it to be slightiy larger than 

I a strict proportion would give it 
The gearing of the minute wheel pinion and the hour wheel is sub- 
mitted to the same law with the same reserves (or the different forms 
of excess which are found in practice. Generally, however, one has 
Minute wheel pinion, division . . . . n +2 
Hour wheel, division, »' -f 2, 




r II 







Lessons in Horology. 

Variotn Calculationi RelalfM b 

290. The preceding table (283') gives a second factor for the 
various gearings which it includes, expressing the radius or total 
diameter of the mobile in function of the radius or primitive 
diameter equal to the unit The use of these factors is impor- 
tant in cases where it is desired to determine the total dimensions 
of the wheels of a gearing by means of a rapid calculation ; for 
example, in the construction of calibres. The solving of the fol- 
lowing problems will rapidly render us familiar with the use of 
this table : 

291. Being given the primitive radius of a wheel to calculate 
its lolal radius R. To solve this question, the number of wheel 
teeth and of pinion leaves in which they should gear must be 
known. One seeks, therefore, in the second column of the table 
for the figures of the teeth range of the gearing, and the corres- 
ponding value indicated in the fifth column will give the factor by 
which the primitive radius of the wheel must be multiplied to obtain 
its total radius. One will, therefore, have 

^ = r X factor of the table. 
If, for example, the primitive radius r of a barrel with 80 teeth 
is 10 mm. and if this wheel ought to gear in a lo-leaf 
pinion, the factor indicated by the table is r.0423 ; therefore, 
^ = 10 X 10423 = 10.423 mm. 
If we had to calculate the total radius of a barrel with 96 
teeth, whose primitive radius should likewise be 10 mm. and gear- 
ing in a i2-leaf pinion, we would have 

.ff = 10 X 1.03904 = 10.3904 mm. 
One sees that the total radius of the latter barrel is a little less 
than that of the lirsL 

292. Being given the total radius of a wheel, to determine its 
primitive radius. This question is the inverse of the preceding 
one and consequendy is solved by dividing the total radius given 
by the tabulated factor. Therefore, 

~ tabulated factor 
Numerically, one has, \l R ^^ 10.423, « ^ 80 and n' = 10, 

Gearings, 203 

293. Being given the primitive radius of a pinion^ to calcu- 
late its total radius and^ reciprocally^ being given the total radius 
of a pinion^ to find its primitive radius. The same as for the 
wheel ; in the first case, one multiplies the primitive radius given 
by the factor of the table, and in the second, one divides the total 
radius by the factor. One has, therefore, 

Rf = r* Y. tabulated factor, 

tabulated factor 

Just as for the proportional compass the table gives two factors 
for each pinion, one suitable for a semi-elliptical excess, the other 
for a semi-circular excess. 

Let, for example, r' = 1.25 and n* = 10, semi-circular form ; 

one will have 

^j. R^ = 1.25 X 1.105 == 1.38025, 

^_i:38o25 _ 

1. 105 * 

If the excess was of semi-elliptical shape, one would have 

R' = 1.25 X 1.15 = 1.4375. 
or, for the inverse problem, 

I. 15 

294. Being given the primitive radius of a wheels one seeks 
for the total radius of the pinion in which it gears (semi-elliptical 
form). One has the proportion 






r — 



which gives the value 

and since 

^/ = r^ X tabulated factor 

one obtains, on replacing r* by its value, 

R' -^r — X tabulated factor. 

Thus, as example, 

r = 5.38. « = 75, «' = 10 

one will have 

R' = 5.38 X ^ X 1. 115 = 0.8249. 

304 Lessons in Horology. 

295. Being given the primHive radtus of a phdon^ one desires 
the total radius of the wheel We have 



'-^ ^ 

^^rxtab. &*. = ,- -^XtaKfac 

Let r* = 0.86, n = 80, x' =: 10. One will write 

R = 0.86 X — X 1.0423 = 7.17. 

296. Being given the total radius of the wheels to find the total 
radius of the pinion. We have 

tab. fac of the wheel 
and ^ «'. 

on replacing, it becomes 

n tab. fac of the wheel 


R' ^ T^ tab. fac of the pinion ; 


„, „ nf tab. fac of the pinion 

n tab. fac of the wheel ' 
Let, for example, 

R = 10.2, « = 80, «^ = 10, 

Tabulated factors i r». . -43 

( Pmion = 1. 15 

one will have 

R' = 10.2 X -^ X -^^- = 1.4067. 

80 1.0423 ' 

29 T. Being given the total radius of the pinion^ to find the 
total radius of the wheel. In an analogous manner to the preced- 
ing case, one will have 

^= ^' 


from whence 

And mtu'<? 
one hrts rtt last 

tab. fac. of the pinion 

r ^ r" 


R' n 

tab. fac, of the pinion nf 

R =z r tab. fac of the wheel, 

n tab. fac of the wheel 

R R' 

n' tab. fac of the pinion 

and since 

one has 

Gearings, 205 

Let R^ = 1.4067, n = 80, n' = 10, one will have 

r> 1.4067 X 80 X 1.0423 

/t = ■■ r-z = 10.2. 

10 X 1. 15 

298. Being given the dista7ice of the centers and the numbers 
of teeth in a gearings to determine the total diameters of the wheel 
and pinion. We know the formulas (185) 

r = D — I — 7-, and r* ■= D \ ,> 

n-\- nf n -\- nf 

R = r tab. fac. of the wheel, 
R^ = r^ tab. fac. of the pinion, 


R = D i > tab. fac. of the wheel, 

n -X- fr 


R^ =^ D 1 7 tab. fac. of the pinion. 

n ~x- n 

D = 5.2, n = 70, n^ = 7, 

Tabulated factors I ^'^^^^ ^^'' ^^^ ^^^^^ 
labulatea lactors | j j^^^ f^r the pinion 

one will have 

, ^ _ 5.2 X 70 X^i.0423 X 2 _ ^ g^^^ 

, ^. = 5j_X7_X 1. 1995 X 2 _ 

77 ^ 

299. Being given the total radius of the wheel and the num- 
bers of teeth of the gearing, one desires to find the distance between 
the centers of the two mobiles. The formula 

r ^ D j — 7 


__ n -^ n^ 

and smce 



.,, t_ tab. fac. wheel 

one will have 

_ ^ n + n^ 

tab. fac. wheel ' n 

Suppose o „ / „ 

^^ R == 4.927, n = 70, n^ = 7, 

Tabulated factors of the wheel, = 1.0423. 

One places _ 4.927 X 77 _ , , 

1.0423 X 70 

300. Being given the total radius R of a pinion and the num- 
bers n and n' of the teeth of the gearing y to determine the distance 
of the centers D, 



In sn analogous 
write the formula 


lessens in Hifrology. 

to the preceding i 

t99S X 7 

301. Being given the diameter P of a walck plate, Ike num- 
iers of teeth n of the barrel and n' of the center pinion, one desires 
b> find : ist, the primitive radii r and r' of the wheel and of the 
pinion .- 3d, the distance of the centers D of the two mobiles ; 
jd, the total radii R and R'. The diameter of the barrel should 
be as large as possible. This question is generally one of the first 
which presents itseSf in connection with the establishing of a new 
watch calibre. 

In order to be able to fit the plate of a watch in its case, a 
"recess" is generally made on the exterior oi this plate, which 
can be valued at one- sixtieth part of the total diameter P. 
There remains, therefore, only ^ available. The useful radius 
b, consequently, 

' 60 120 

The extremity of the teeth range in the barrel can coincide 
with the extremity of this radius, the teeth finding the necessary 
play in the hollowed-out part in the center of the case. 

The radius -j-Vff ^ should be equal to the sum of the following 
lengths : the primitive radius r" of the center pinion, the primitive 
ndius of the barrel and its total radius (Fig. 81). 

One would, therefore, have 

one will also have 



from whence one finds 

r = 



-|- I -f tab. fac. 

Knowing r, it is easy to determine r', since 

r'= r 


the distance of the centers D will afterward be determined by the 
sum of the two primitive radii. 

Fig. 81 

Knowing r and r', one will calculate the total radii R and R! 

by the operation : 

R = r tab. fac. of the wheel. 
R* z= f' tab. fac. of the pinion. 

Suppose, as an example, /* = 43 mm. (19 lines), « = 80 and 

n' = 10. 

Tabulated factor of the wheel = 1.0423. 
Tabulated factor of the pinion, = 1.15. 

One will have : 


r = 



i + I + 1.0423 

2o8 Lessons in Horology, 

or also 

It follows that 

and finally 

Since one has 

^^ 59 X 43 ^ 59X43 

lao (i + I + 1.0423) 260.086 

r = 9-7545. 

D = 9.7545 + 1.2193 = 10.9738, 

R = 9-7545 X 1.0423 = 10.17. 
^ = 1. 2193 X 1. 15 = 1.4- 

^-43 = 21.1416. 

one should have likewise, or very nearly, 

r^ -]- r + /^ = 21.1416. 
The addition gives 

1. 2193 



With this approximation the result is satiskctory. 

302. Being given the total radius R of the pinion in which 
the rack of a repeater gears^ the number n* of teeth cu:cording to 
which the pinion is divided^ and the total radius R of the rocky 
one desires to find the number according to which the sector of this 
last xcheel must be cut. 

Let us admit the ogive of the teeth range of the two mobiles 
equal to twice the diametrical pitch (277). We will have the 
primitive radius of the pinion by the formula 

and that of the wheel 

r^ = 

r = 

R' n" 

w' + 2 

R n 

Sii\oo one has 


T ^ 


- ~» 

one can also write 




ir' -f 2 

R' n" {n + 2) 


A^ n 

/? i» (i/+ 2) 

ft ' 1 

*f>n<^ Vn<^wit X\\aX th« diviftton of this pinion br ih^ siet of teeth is not complete, for the 
«>Ni»iM( thAt thin moMl<» oul^r 4>x<khU«« « f^Actlon of « turn. The pitch of the gearing left fUI 
tllotUtiitM (ho Mrix>»t of U)<» woT^m^pnU 

n simplifying, 

R («' + 2) = /" (» + 2). 
I One finally obtains 


Suppose, for example, R ^ 9' 961 R' = 
would have : 

« = 9 9^ X ^5 _ _ , _ ft, ,„,u 

Remark.— One could obtain directly the above formula by a 
iropordon analogous to that of 289, 

303. The following problem does not find its solution in the 
theory of gearings only, but also in that of trains and of the 
motive force. It recalls to our mind, in an excellent manner, the 

P Studies that we have gone over, so we do not hesitate to close this 
■eries of problems by joining together some of the various ques- 
tions which we have treated in this chapter and in those which have 
preceded it. 

304. Problem. A horologist has constructed a stem-winding 
watch the diameter of whose plate is 50 millimeters (22 lines). 

^H Upon winding the watch, he notices that the power necessary to 

^^koperate the winding works, that is to say, to overcome the force 

^^■lOpposed by the spring to the movement, is too great. He decides 

^^B'then to manufacture a new watch, like the first, but in which the 

^^^winding can be more easily eflected. He should, therefore, modify 

^^Vthe value of the two factors which enter into the expression of the 

mechanical work ; the force on the one hand and the iime employed 

for the winding on the other ; in other words, the spm:e traversed 

by the point of application of the active force (37). 

»The first watch has its barrel furnished with a stop work of 
4 turns, and runs for 32 hours ; the second should run for the same 
number of hours. It is evident that if we introduce into the 
second watch a barrel furnished with a stop work, allowing it to 
make 5 rotations during 32 hours, we wilf have diminished the 
average tension of the force and augmented at the same time the 
^H,duration of the winding. 


Let us seek, therefore, lor the nature and the value c 
change that must be made in the second construction, in order to \ 
arrive at the end desired. 

SOS. In the first place, the relation between the numbers of teeth ] 
in the barrel and of leaves in the center pinion must be changed. 

In the first watch, this relation was \^, which gave a length erf J 

-- X 4 ^ 3* hours. 

In the second watch, one should also have 

-^- X 5 = 3= hours. 

On choosing a pinion of lo leaves, one will have 
K = 6.4 X lo = 64 teeth, 
or a pinion oi 15 leaves, 

n = 6.4 X 15 = 96 teeth. 
It is not an absolute necessity, in general, to conserve 
above relation in a very strict manner. Thus, if one wished a 
pinion with 12 leaves, one would have 

n = 6.4 X !2 = 76.8 teelh. 
This fractional number not being practical, let u 
example, « = 78 

We would then have 

'- X 5 = 325 hours. 

The watch would run, with the above number, half an hour 
longer than was desired. 

Since, for a watch of 50 mm. diameter, a barrel with 96 teeth 
does not give too weak a teeth range, one can accept for this 

96 teeth for the barrel, 

15 leaves for the center pinion. 

306. The primitive radii of the two mobiles must now be cal- 
culated and the distance between their centers. Let us commence 
by seeking for these values in the first watch, in order to compare 
the results, 

Gearings. 211 

The formula which we have determined (301) gives us 

+ I + 1.039 


and making the calculation ^ ^ 11.36. 

Then . «^ r v^ 12 

r' = r — - = T1.36 X -7- = 1.42, 
and n 96 

D = 11.36 + 1.42 = 12.78. 
The total radii of the two mobiles were : 

R = 11.36 X 1.039 = 11.803, 
R =^ 1.42 X 1. 14 = 1. 619. 

For the second watch, the calculations are naturally analogous, 

only we have here the case of a gearing whose teeth range is not 

found in the table. 

One can admit, in this case, by analogy the tabulated factor of 

^he wheel equal to 1.04 by slightly forcing the figure of the factor 

"^ the preceding gearing, since the height of the ogive should 

'*^ crease with a larger pinion (15 leaves instead of 12). 

For the pinion, we will concede an excess with semi-circular 

^*Xape perfectly admissible for this number of leaves. 

We will thus have 


r = 


X 50 

^-nd making the calculation 

^ + 1 + 1.04 

r = 11.17; 

and , 15 

-Therefore, ^ 

D = 11.17 + 1.745 ■= 12.915 

ADne will have the total radii 

R = II. 17 X 1.04 = II. 617 

ig^ = r^4- ^ \f = r^ (i -\- -^^ = 1.891. 
15 X 5 \ 37.5/ ^ 

One sees that the diameter of the barrel has diminished and that 
of the pinion and the distance between the centers has increased. 

30t* Let us now seek for the exterior and interior radii of the 
barrel drum. 

These dimensions ought to be as large as possible. The ex- 
terior radius of the drum in the first watch was 1 1 mm. , therefore 


Lessom in Horology. 


That of the second i 

0.36 less than the primitive 1 
could, therefore, be 

11.17 — 0,36 = lo-Si mm. 

The Ulterior radius of the drum was, in the first case, 
11 — 0.77 = 10.33; 
consequenUy, it could be, in the second case, 

10.3I — 0.77 = 10.04 or 10 mm. 

308. Let us now calculate the dimensions of the hub, and thc'^ 
dimensions as well as iheyurce of the spring. 

In the first watch, when the spring is pressed against the side 
of the drum, it occupies the third of the interior radius of the 
barrel, another third remains empty and the third third is occupied ■ 
by the hub. This spring makes, in this position, 15 turns and 4.5,'! 
placed loosely on a table : the number of turns of development is 6.5J 

The dimensions of the blade are the following : 

Thickness ^0.21 
Height ^3.9 
Length = 780 n 

According to the formula (97) 

^ E h e' 


the moment of the force of this spring if 
F = 557-64- 
When the spring is coiled up, that 
is completely wound, one has 

>i = 15 + 6 — 4.5 = 16.5; 
consequently, p ^ ,^^ y, ^^^ _ g,^ g,. 

When the spring is down, one has n' = IZ.5 and 

F' = 12.5 X 557 = 6942 gr- 
in order that the center wheel may receive in the second watdlJ 
the same force at the beginning and at the end of the tension of t 
spring, it is necessary that, the watch being wound, one should h 

^1' = -|- X 9190 = 7352 gr- 
and if the watch is run down 

■'^i' = -7- X 6942 = S553 er-. 

Gearings, 213 

since for the same number of teeth in the barrels the relation of the 
leaves and pinions is ^^ . 

^ "~ T' 

In order that this difference between the moments of extreme 
force may exist with 5 turns of the stop work, one must have the 
proportion ^6^ ^ «/ + 5 

12.5 «/ ' 

from whence one finds , , 

«/ = 15625, 

and afterwards 

«i = 20.625 — «. 

309. Let us further calculate the thickness of the spring. Let e 
be this thickness for the spring in the first watch and ef that of the 
spring in the second. 

For the first case we have the moment 

„ E h e* 2 11 n 
^= 12 L 
and for the second 

4 ^ £ A e^* awjn 

on remarking that the length L increases in inverse relation to the 
diminution of the thickness e. 
One could, therefore, place 

4 £ h e' 2wn £ Ae^' 2ir ^ n 

"5 12^ " 12 L. S ' 

from whence, on simplifying, one obtains 

5 ^a 



from whence 

5 4 

25 ^* = 16 e\ 

\ 25 \ 5 

For e = 0.21, one has 

e^ = 0.21 X 0894427 = 0.10826. 
310. Since we have the proportion 

L e" 

'214 Lessons in Horology. 

one could iinall]r calculate the new length of the spring on placing^ 

L' = 871 mm. 

Since the second barrel is a little smaller than the first, this 
spring will fill it a little more than one-third ; but as it is also 
thinner, one can diminish the hub proportionately to the relative 
thickness of the spring (m). 

Thus, the first hub having a radius equal to one-third of the 
interior radius of the barrel, this one would be -y- ^= 3.66 . . mm. 

This radius being 17.777 times greater than the thickness of 
the spring, the second hub could l>e 

17.777 X .18826 = 3.34.. .mm. 
31l> We know, moreover, the means of increasing the ease oi 
the winding by increasing the number of teeth in the ratchet and of 
the contrate teeth range in the crown wheel ; since we have already 
treated this question (169), we will not go back to it here and we 
will thus admit that the problem proposed is solved. 

Conical G««rin^. 

312. In the gearings that we have just considered, the two axes 
are parallel to each other and we know that the movement of the 
system can be compared to that of two cylviders mutually conduct- 
ing each other by simple contact We have designated gearings oi 
this sort under the name of cylindrical gearings. 

313. If, in place of being parallel, the two axes are concurrent, 
one can imagine thai the movement of one produces the movement 
of the other by the contact of two cones concentric to each of the 
two axes (Fig. 82). This system takes, therefore, the name of 
conical gearing. 

The two axes can form any angle whatever with each other ; 
wc will treat particularly the special case where this angle is a right 
angle, almost the only case in horology. 

Suppose (Fig. 82) O x and O y, two perpendicular axes 
iiround which turn the two cones COB and A O C\ let us admit 
thnt their movement is produced without slipping. 



As for the cylindrical gearings, one can prove th'it the rela- 
tion of the angles traversed by the two cones is inversely propor- 
tional to that of the diameters C B and A C 

The diameters can be measured in any manner whatever, pro- 
vided that their circumferences be tangent. Thus, in place ol C B 
and A C one can just as well take C B* and A' C\ since these 
straight lines form the sides of similar triangles. 


Fiff. 83 

» and «' being the angles traversed in the same time by the 
two cones, one has, therefore, the proportion 

A C ^ A ' C __ a' 
C B ~ C B' ~ \' 

and since, n and n! being the numbers of teeth 

one will also have 




A C 


C B n' 

314. The pitch p of the gearing varies with the distance O C; 
for such a point of contact C, it is 





lessons in Horology. 

315. Form of the Teetll. As in the cylindrical gearings, the* 
transmission of the movement cannot be effected practically by the 
simple contact of the two primitive cones ; one is generally obliged 
to supply these cones with flanges, that is to say, with teeth, which 
malce them move as if they were driven by their simple adhesion. 

The contact, consequently, is not always found on the line O C; 
if we consider the contact at the point C, the displacement of this 
contact should take place on the surface of a sphere whose center 
is at O and passing through the points A C B. The form of the j 
teeth must be traced on this sphere. 

Thus, M O and O N (Fig. 83) being the two axes of rotation 1 
which meet in a point O, let us take this point as center of a | 
sphere ; it will contain the two upright cones, as we have just said, 
having their common summit at the center O, and will cut them I 
■long two circumferences, of their bases, tangent at the point A I 
belonging to the generatrix of contact of the two cones. 

These circumferences drive each other, exactly as would the 1 
primitive circumferences of a cylindrical gearing situated ir 

ae plane. The sphere playing the role of the plane consideredj 



in the first case, all the p ropert i es aheady determined are here 

Thus, one can determine the cnnre described by a point of 
one of these primitive drcnmferences by making one of the cones 
roll on the other which remains immovaUe ; the curve thus a^es- 
dered is the spherical epicycUnd B C. 

On account of the similarity of the methods employed in this 
case with those that we have previously described, we will not enter 
into all the details of these constructions. 

116. Let us examine, howe\'er, the case cA JUmk gearings. 
The flank being a diametrical f^ane of the primitive cone, the drivii^ 
tooth will be a conical surface whose form must be determined. 

Suppose S O and S (J the axes of the two primitive cones 
which should turn while touching along an ec%e S M ^not repre- 
sented in the Fig. 84 ^ Suppose Mm NzdA Mn^ N" the circmn- 

ferences of Ihe 
cones by Ihe 

proceetiiag fe-ocn the rntersectioo of 
drawn pterpetwixcnlarly throcgfi rfie 

On die radios J/ <7 or due cnrcie 
a dicassS/txfsat CT zodi ihcuughics 



Lessons in Horology. 
1 its plai 

; this perpendicular i 

C one erects a perpendicular o 
meet the axis at a point S. 

II one considers this point 2 as the common summit of twaa 
cones having for bases the two circles O and O", and if one make9l 
the second cone S O" roll on the first S <3, a point of the < 
ierence (?" will describe a curve m m" , a spherical epicyclaHA 
situated on the sphere on which is moved the circle O" 
sphere having its center at S. 

If one made a cone which had its summit at the point ^ p 
through this epicycloid, this cone will be the exterior surface of a 
diametrical plane of the cone ^ O and should consequently be 
taken for the surface of the cone ,^ O. This result appears evident 
from the similarity in the construction of cylindrical gearings, there- 
fore, we will add no other 
proof to the application 
of this development by 

3ir. Besides the 
cycloidal iorm, one 
ploys also the evolvent ol 
circle for the teeth of coni- 
cal gearings. 

318. Construction fA 
Conical Gearins's. By the 
preceding, all the lines 
which enter into conical 
gearings being defined, it 
is only necessary to apply 
the principles of descrip- 
tive geometry to deduce 
\. y irom them the outlines nec- 

-^ .— '^ essary for its construction. 

Fly. 8s But it is useless to enter 

into extended details wilJI' 

regards to this, since in practice a more simple method 

been adopted and one sufficiently exact. 

In order to represent these forms in a more convenient 
manner on paper, one substitutes for the spherical surface the 
plane surface projected zX F C G (Fig. 85) perpendicular to the 
line of tangency of the primitive cones O C. On this surface 


Gearings. 219 

one represents the developed surfaces of the two cones A G C 

and B FC{¥\g. 86). 

Ill tlie development of the cone O D, the circumference prxj- 
jecled in A D C will become an incomplete circle A C A with 
radius .^ C of the same length as 
this circumference. Likewise, the 
cone O E developed will give an 
arc of circle of same length as the 
circumference projected in C E B. 

It is on these circumferences 
that one lays oft the lengths cor- 
responding to the pitch of the 
gearing determined on the circum- 
ferences with radii A D and C EkA 
the bases of the two cones. One 
draws afterward the form of the 
teeth of the two mobiles, as one 
does it on the primitive circum- 
ferences in the cylindrical gearings. 

These forins are obviously 
equal to those that one would ob- 
tain on the spherical surfaces them- 
selves, since, for the small dimensions of a tooth, the surface of 
the plane and that of the sphere are almost the same. 

For the purpose of being able to compare the form of the 
teeth of a conical gearing with that of the drawing, one terminates 
these wheels by a portion of the cones C G A and C5 /^(Fig. 85). 

319. In horology, however, one cannot do this, either on 
account of the lack of room or because the wheel carries at the 
same time another teeth range, as the crown wheel of stem-winding 
gearings, for example. The exterior surfaces o( the two wheels 
are then straight planes, perpendicular to the axes. 

Admit, for this case, that the exterior planes of the two 
mobiles meet in C(Fig. 87), and let us seek for the form of the 
teeth cut by the plane C B. It is, in fact, on this plane that 
we see the form of the teeth range and that we can determine 
its dimensions. 

Let us first draw the two primitive cones COB and A O C. 
the latter being represented only by its half DOC; draw the 

mdicular F Gxq Cand a parallel F' G' to F G, finally the 


Lessons in Horology. 

primitive circumferences with radii F' C and G' C* and determine 
the form of the teeth according to the method known. 

In order to obtain a horizontal projection of the tooth of the 
wheel (winding pinion, for example) whose center we can place at 




• % 

\ : 

% 1 


i— •—•?"•-* 


"• 1 



*^ ' 


. v^ 


v» Jm y y -' 

^*^ . fen !'^.'»*. -' -V^ . ' « \\ • 

• • '•'.♦ V *'-y^ ' \ 


. ■■■'i 



()", let lis note that the point C is projected at Cand C", we will 
lmvi% therefore, the circumference of the primitive base of the cone 
wllli mdiuH E C •-- O" C"; let us lay of! half the thickness of the 
Ippih C il* on each side of C" on the primitive drcle. 



In order to determine now the total radius (7' /", correspond- 
ing to the point /', let us project this point /' on the plane F G 
at /; draw the radii O /prolonged to M\ the point of intersection 
M of this radius with the plane B C gives the total radius E M 
that one can project on O* /"; from the center O* describe after- 
wards the total circumference of the wheel. 

One proceeds in an analogous manner to determine the bottom 
of the teeth on the plane B C hy projecting the point K' on F G, 
drawing the radius O K cutting the plane ^ Cat iVand projecting 
this point at K" ; one will have thus the radius O K'* of the cir- 
cumference passing through the base of the teeth. 

In order to determine finally any point /*" of the form of the 
teeth, project the point P' to P on F (7, draw the radius O P U 
and project the points P U on (9" /". From the center (7' one 
causes to pass through the points obtained arcs of a circle and lays off 
the half thickness ^S*' P' on the circle passing through ^S*"; and 

322 Lesiont m Horology. 

draws, afterwards, the radius O" T". The fx^t P" at which this 
radius just cuts the circle projected from U, is the point of the tooth. 

Let us stitl seek for the form of the teeth which appears on the 
interior plane L Q parallel to the plane B C M {Fig. 88). 

On the side elevation, the point Q represents the point of the 
teeth. Project this point on the front elevation, at Q", and describe 
from O" as center the circumference which passes through this point 
and which gives us the point of the teeth. To obtain the point T" 
of the base of ihe teeth, draw the radius O N, cutting the plane 
L Q 3.1 T and project this point on the front elevation, we will 
thus obtain the point T" through wliich one passes the circum- | 
ference of the base of the teeth. I 

In order to further obtain any points whatever, for example, 
those which are found on the primitive cone, one draws the radius 
O C, cutting the plane I. Q aX. V, projects this point on the front 
elevation, at V" and describes the circumference from the center O". 
One afterwards draws the radius O" R ; the points determined by 1 
the intersection of this radius with the circumference passing J 
through V", are points of the curve of the teeth. One can thus I 
determine as many points as one desires and represent in thiaf 
manner the complete form of the tooth. 

Let us remark that, compared with the form determined for 
the teeth on the plane F G (Fig. 87), the form obtained on the 
front elevation having 0" as center, is elongated. One should, 
therefore, take account, in practice, of the elongated shape of the I 
teeth in these wheels compared with those of corresponding planefl 

The drawing of the front elevation of the wheel is made i 
exactly the same manner. 

Defedi which PrMent Themielves in theie GcBrin^i. 

320. When, in a gearing, the normal to the point of conta 
does not pass through the point of tangency of the primitive cii^J 
cumfercncos, the transmission of the force is irr^ular. 

The faults of construction which most often produce 
effect, are : 

First — A relative disproportion between the total diameters d 
the two wheels. 

Second — A gearing too close or too slack. 

Third — A bad te*;th range. 



According to the case, one will then find in the gearing a 

' ' butting " or a " drop. ' ' 

321. The butting, also called binding, is the irregular contact 
of two teeth before the line oi centers. If, for example, a is the 
point of tangency of two 

primitive circumferences O 0| 

and ff (Fig. 89) and c the /'"' \ ""•■■._ 

point of contact of a tooth 
and a leaf, one will find on 
drawing the normal to this 
point that in place of pass- 
ing through a it will cut the 
line of centers at a point a' 
situated between a and O . 
There will result a diminu- 
tion of force transmitted at 
this instant for the two fol- 
lowing reasons : 

First — In place of a 
force F' ^ -^ §^i ^"^ *''" 
have only F' ^= F -^^, . as 
much different from the first 
as the point a' is found 
nearer to the center C. 

Second— Increase of 
the re-entering friction. 

The causes which can produce this defect a 

(i) Two slack a gearing; 

(2) A pinion proportionally too large ; 

(3) A bad teeth range. 
Fig. 89* shows the case of too large a pinion ; the pitch of 

the gearing is longer than that of the wheel. The tooth B has 
ised to conduct the leaf and the tooth A enters too soon into 
contact with the succeeding one. As we have said, the moment 
of the force transmitted is, therefore, diminished. 

Fig. 90 shows the case of too slack a gearing. In place of 
entering into contact with the straight flank of the leaf, the tooth 


e generally : 

le tfae dtfecti thai tfaef 

d be uf teai-eUlpticBi to 

^ Bppreclflbie for piuli 

conducts, first, the excess, the normal cuts the line of centers be- 

the center O oi the pinion, 
the case oi a bad teeth range ; the tooth, too 
short, for example, has 
its contact with the 
leaf, as in the preced- 
" ing case : the normal 
passes between the 



point a and the center 
O and one has a dimi- 
nution of the force 

When the above 
defects are not too 
much accentuated, it is 
possible to remedy 
them, in order to ob- 
tain a passable g'ear- 
ing ; but, at least in 
the first case, it is im- 
possible to arrive at 
absolute perfection, 
If the pinion is 

Fig. 90 slighdy too large, one 

can touch up the wheel 
in such a manner as to free the teeth range at the base a b (Fig. 92) 
and make it less pointed, after the manner of the English teeth 

If the gearing is too slack, one increases the diameter of the 
wheel by careful forging. 

If the teeth range is defective, one can try to rectify it by means 
of a suitable ordinary cutter, or, still better, with an Ingold cutler. 
322. II the first contact of the tooth with the leaf com- 
mences after the line of centers, it may happen that at a certain 
moment of the movement the angular speed of the wheel becomes 
proportionately greater than that of the pinion which it conducts. 
This defect is the drop ; it is produced by 
(i) Too close a gearing. 

(2) A pinion proportionally too small. 

(3) A bad teeth range. 



Fig. 93 shows too small a pinion ; the pitch of the gearing of 
the wheel is greater than that of the pinion. When the tooth B 
should cease the contact on the generating circle, the tooth A is 
still found removed from the leaf that it should conduct. The 
tooth B will slip along the flank of the leaf and at this instant the 
normal to the point of contact will not pass through the point 
of contact of the primitive circumferences, but will cross the line of 
centers at a point nearer 
the center of the wheel ! 

One will, therefore, \ 

have, in this case, an ^ 

increase of the force 
transmitted. For a 
uniform movement of 
the pinion, the wheel 
will take an accele- 
rated movement ; this 
is, technically speak- 
ing, a * * drop. * ' 

Fig. 94 represents 
too deep a gearing, the 
tooth B conducts its 
leaf farther than the 
generating circumfer- 
ence ; there is, there- 
fore, produced a slip- 
ping of the point of 
the tooth against the 
flank of the leaf, the accelerated movement which the wheel 
takes terminates by a drop of the tooth which follows on the 
leaf which it will conduct. The direction of the normal at the 
point of contact shows that one has, in this case, also an 
increase of the force transmitted. 

Fig. 95 represents the case of a bad teeth 
range of the wheel. The teeth, which are too 
long, drive the pinion leaves farther than they 
should geometrically ; one can thus recognize 
the drop which will be produced. 

A gearing presenting the above defects can 
Fig. 93 be corrected by diminishing the height of the 

Fig. 91 

226 Lessons in Horology. 

ogive in such a manner that the teeth drive the leaves a less d 
tance or, otherwise, by forming the teeth in such a manner as 
give them a greater breadth on the primitive circumference. 

323. Oi 

of a wheel 


just done, that the gearii^ 
pinion produces a butting, that, 

the other hand, too deep a gear^ 

j ing produces a drop, one i 

j that it is best to make a de^ 

gearing when the pinion is t 


Reciprocally, a gearing who9( 
pinion is too small should 1 
relatively shallow. 

324. A defect which oni 
encounters often enough in gear- 
ings is that which is 
by pinions whose lee 
long enough, that is to say, 
pinions which are not cut t 
! enough. If the teeth of the 

wheel are correct, one finds very 
0* often the point of the tooth i 

Fig. i);i contact with the bottom of the 

leaves (core of the pinion), 
one cannot change the pinion, which is the only means to obtain a 
perfect gearing, the ogives of the teeth must then be shortened^ 
either by cutting off the points or by modifying the shape. One 
understands that in these cases absolute perfection exists no longof, 
especially if the number of pinion leaves is small ; since then thu 
contact should commence before the line of centers. 

325. One encounters very often, also, pinions of ordinar] 
quality in which the flanks of the leaves are not directed toward 
the center, but are diverted more or less from it Such piniooE 
should be rejected as much as possible if one wishes to preserW 
in the gearing the quality of a flank gearing ; if not, the tooth a 
the wheel would have to be formed by means of a curve describel 
as we have indicated (215). 

In a gearing, detective either on account of the shape or direo 
tion of the pinion leaves or the wheel teeth, if one modifies one a 
the two profiles it might happen that one arrives at a coned 


gearing fullillmg all the con- 
ditions of a uniiorm trans- i 
mission of the force, even \ 
when the essential character- ^ 
btics of the flank or epicy- 
doidal gearing no longer exbt. 
In this case, the entire theory 
of the determination of the . 
forms of contact is there in 
order to make us understand 
that one has luckily been able 
to find a combination of forms 
fulfilling the condition estab- 
lished, that the normal to the 
successive points of contact 
passes constantly through the 
point of tangency oi the primi- 
tive circumferences. We know 
that this condition suffices for 
the gearing to be perfect, 
whatever may be the shape of the profiles established. 

Paiiive R«iuluic«t In Gearing. 
326. General Ideas. We have already indicated that the pas- 
fflve resistances are forces which naturally present themselves in 
all machines in motion(46). 
2J These resistances are of 

diverse natures : some pro- 
ceed from the bodies them- 
selves, from their weight, 
their form, their dimen- 
sions, and also from the 
relativeness of the move- 
ments which animate them. 
Such are friction, and its 
congenerics, inertia and 
shocks. Others arise, more 
properly, from the medium 
in which these bodies are 
moved, such as, especially, 


338 I^ssffns in Horology. 

the resistance of the air. Among tliese, the principal cause c 
absorption of work which is to be considered here is the friction., I 
of which we will first take up the general study before applying J 
the laws to the particular case of the gearings. 

327. When a body is moved by slipping on another body, 
there is produced a resistance which is opposed to the movement. 
This resistance is due to the action of the two surfaces in contact, 
when the movement already communicated to the body allows the 
inertia to be excluded. This resisting force is friclion ; it appears 
to proceed from the reciprocal action of the molecules of the two 
bodies. The inequalities of surface more or less evident in thesej 
bodies penetrate each other reciprocally, fit into each other witha 
much greater intensity in proportion as the two bodies are more* 
closely pressed together. Moreover, when one of these bodies ^a\ 
displaced the resistance produced by this "binding" is hirtherl 
increased by the driving back ol the molecules situated in front of 1 
the moving body, 

328. Besides this cause of resistance, there exists a second one^B 
due to the adhesion of the two surfaces. The effect produced 1 
this second cause can be made very apparent by placing on eacJi:l 
other two planes of the same kind ; if the surfaces are very care-J 
fully planed and perfectly polished, as, for example, those of two J 
mirrors, the adhesion may become so great that the separation rfj 
the two bodies becomes very difficult.* 

Friction depends, therefore, on the two causes mentioned ; but^l 
the last is very often neglected if the two surfaces are directly in I 
contact, that is to say, if there is no coating or lubricating sub- \ 
stance, such as oil, between these surfaces. Numerous experi-j 
mcnts have, in fact, proven that this resistance may be neglected! 
when the extent of the surfaces in contact is not very great. 

329. But when one interposes a greasy substance betweeni 
the two bodies, it is no longer possible to neglect this last cause, I 
which, in certain cases, may diminish the friction properly ■ 
speaking and, in others, increase it. We will treat, further « 
of this question and will hmit ourselves for the moment to th«i| 
study ol "dry friction." 

• Thl« pbonnmeiion arlaea from Ihe mora or lew mmpleta eipulaloii of [he air l)Mire««.f 

Gearings. 229 

330. The Two Kinds of Friction. If the same part of the 
surface of one of tfie rubbing bodies always remains in contact 
with the other ■ body, there is shding, and the friction takes the 
name of "sliding friction." If, on the contrary, the surfaces in 
contact change at each instant, there is rolling and the friction 
takes the name of "rolling friction." An example of the first 
case is the friction which is established during the movement of a 
sleigh along a road ; and of the second case, that which is pro- 
duced when a wheel rolls on a plane. 

^We will occupy ourselves especially with the sliding friction, 
only kind which we will encounter in horology. 
331. The slipping may be linear, that is to say, be efiected 
aiong a plane or any surface H-hatever when one of the bodies is 
continually displaced with relation to the other ; or it may be 
circular, if one of these bodies turns on itself without going 
forward, for example, a trunnion in its bearings. 

The friction of the teeth of a gearing is produced by a linear 
slipping ; that of the pivots of these same wheels in the interior of 
the holes in which they turn is produced by a circular slipping. 

332. Laws of Friction. It has been discovered by very care- 
ful experiments that the resistance due to friction is subject to three 
principal laws which can guide in the applications and which are 
sufficiently exact within the limits between which they are con- 
sidered in machines. 

^^m Vivsl— The friction is proportional to lite normal pressure ; 

^Hftiat is to say, the resistance is always the same fraction of the pres- 

^^^Bre which applies one body on another, which is easily understood, 

^^Bbce the actions of the molecules should arise by reason of this 


^^V Second — The friction is independent of the surfaces in contact; 

^^This is to say, when this extent increases without the pressure 
changing, the total resistance remains the same, although the pres- 
sure on each element of surface is found to be diminished in inverse 
relation to the extent of these surfaces. Since, for given sub- 
stances, the friction is a constant fraction of the pressure, it fol- 
lows that a heavy body drawn on a plane gives rise always to 
the same resistance, whatever may be the extent of the surface 
of contact. 

Third — 7^ friction is independent of the speed of the move - 

^^^ment; which is to say, that the same amount of work is necessary 


Lessons in Horology. 

in overcoming the iriction o( a body traversing a certain distance, 
no matter what may be the speed which animates the body. 

By the aid of these three fundamental laws and of the values 
determined, experimentally, in order to establish the relation of the 
friction to the pressure according to the nature of the surfaces in 
contact, one may value in each case the work absorbed by friction. 

333. Experimental Determination of the Force of FricliOD. Let 
U9 mippose that a body with weight P be acted upon by a force F 
which makes it slide with a uniform movement on a surface A B 
(Fig, 96). Ont' knows that when a body is moved uniformly, 


lig. I 

the algebraic sum of the forces which act on this body is equal to 
zero. The force F should, therefore, be equal, and in contrary 
senHC to, the force of friction : it will be, therefore, the measure of 
the greatness of this resistance. 

One of the laws which we have cited, showing that the friction 
is proportional to the pressure, it follows that if the weight P is 
^_ doubled, the friction is doubled at the same time and consequendy 
^H its equivulent F. 
^H The relation —^ is, therefore, constant for the same substances 

^H in contact : this is called the "coefficient of hiction," that is gen- 
^H erklly represented in the calculations by the letter/. Thus one has 


When this coefficient is known, as well as the pressure P 
extended nonnally to the surfaces in contact, one can determine the 
Irictioa F by multiplying the pressure P by the coefficient / 
Tlwiirfiwe, F = f p. 

3M. Let us note that the co^Bcient -A friction does Dot always 
keep tbe sune wKw tor diffcnnt surttces <A the same knid. for tke 

Gearings. 231 

r a body is and the more it is polished, the less is the friction. 
Its value is also modilied by interposing a greasy substance, oil, 
for example, between the surfaces in contact. The object of this 
operation is principally to avoid the grating and the heating of the 
frictioning bodies. One knows, in fact, that without this precaution 
there are detached from the surfaces small fragments which groove 
them deeper and deeper ; the friction speedily increases and the 
heating which results from it can even go so far as to make the 
bodies red hot and to set them on fire if they are combustible. 

One finds that friction of steel on steel produces by the grating 
a reddish dust, which is, probably, oxide of iron ; the dry friction 
of steel on brass enables us to prove that a certain quantity of brass 
is deposited on the surface of the steel ; the heating should, in this 
case, be considerable. 

Horologists know the grooves, often very deep, which the lack 
of oil on the pivots produces, when these turn a long time, dry in 
their holes (the fact is especially noticeable on the pivots of the center 
wheel) ; they are familiar also with the deep lines worn in the 
leaves of tempered steel pinions, caused by the teeth of the wheels 
made of gold alloys, which, for this reason, are almost entirely 
abandoned in these days. One sees by these examples that a high 
speed of the mobiles is not necessary to produce the grating, 
which is on the whole entirely in conformity with the third law of 

335. The following table is intended to give an idea of the 
mean value of the coefficient / in the most general conditions. It 
is best, in each particular case, to choose this value properly, 
according to the probable conditions of the action of the parts in 



Metal on metal 

Metal on precious stones .... 

Wood on wood 

Bricks and stones on the same . . 
Leather tiands on metallic pulleys . 

CIS to 0.17 



0.30 to 0.40 


Lessons in Horology. 

In large machines whose frictioning parts are carefully greased 

the coefficient of friction diminishes to a value of / = 0.08. 

336. Work of Friction. The mechanical work of a force being 
the product of this force by the path traversed by its point of 
application, when the path and the force have the same direction, 
one will have, if E is the path traversed, 
W. }^ ^f P E. 
If the two bodies are movers, it will be necessary to consider 
the two forces of friction which have / P for common value and 
which act on each body in the inverse direction of its movement 
with regard to the other, each of the forces producing work. 

Suppose that the movements of the two bodies A and B 
(Fig. 97) are effected in the direction of the arrows (i), ^and E' 
being the respec- 
tive paths traversed 
in this direction, 
which will be that 
of the relative 
movement of the ^ 
mpm body A, if E i 
greater than E'. 

If we consido! 
the movement 
the body A the friction will produce a work / P. E., which 1 
be negative, since the direction_/'/'is the inverse of that of £■; t 
is, therefore, a resisting work. On the contrary, the work of j5 o 
A will produce on this first body a positive work y /". E', 
will take away from the resisting work /P. E,, so that finally th« 
resisting work produced by the friction will be 

W. F --=^ f P E — / r. E' = / P {E — E'), 
E — E' being positive. If E— E' becomes negative, the equa-^ 
tion no longer holds, and the direction of the friction must 1 
changed ; in place oi having f P (£ — E'), one will have 

Let us here note that the work developed by the friction ( 
one of the two bodies is positive ; with regard to this body, there-^ 
fore, friction plays the part of motive force. This property ii 
employed industrially in the transmission of movement by cylin- J 
ders, cones or friction plates. 

Gearings. 233 

33?. Anfle of Friction. Suppose a body resting on an inclined 

plane A C (Fig. 98) ; let us admit that we have regfuialed the 
inclination oi this plane in such a manner that the body may be on 
the point of moving, or, what amounts to the same thing, that it is 
moved with a uniform motion, the length of this plane. In this 
case the force of the friction is equal and in a contrary direction to 
the force which acts to make it descend. 

^H The weight p of the body acts along a vertical line e a, pass- 
^mg through its center of gravity ; making c a ^ p, drawing the 
line c b parallel to the plane and c d perpendicular to this direction 
one will be able to form the parallelogram of the focus by drawing 
the lines a d parallel to ^ 1: and a b parallel to c d. The length c b 
will then represent the value of the force F tending to make the 
body descend along the plane, and the length, c d, the normal 
I pressure P. We will, therefore, have 

FIk. S8 


: and B C A give, moreover, the 

^ = — =/. 

The similar triangles & 
proportion ^ ^ 

Vi ^ 

One can thus see that the coeilicient of friction is equal to the 
quotient of the height B C oi the inclined plane divided by the 
length of the base A B. 

338. Designating by +, the angle, CAB, that the incHned 
plane forms with the horizon when the movement takes place, the 
two components of the weight p can also be represented by 

p. sin + -. f- 
following the direction of the inclined plane and by 


Lessons in Horology. 

perpendicularly to this plane. One has, therefore, 
F / sin + . . , 

One discovers, on varying arbitrarily the extent of the 
in contact and the weights of the bodies, that the angle of inclii 
tion does not vary for the same substances in contact 

This angle is called the angle of friction and the numerical 
value of the relation of the friction to the pressure, equal to tang. ^, 
is the coefficient of the friction. For hard and polished metals and 
the stones used in horology, this angle has a value varying front 
7" to 8" 30'. 

Example of Application. — On a plate of tempered and 
polished steel we place a ruby, a lever pallet, for example. We 
elevate little by litde one of the extremities of the plate until the 
ruby commences to slide with a uniform movement. The height 
B C Xo which it was necessary to elevate the steel plate being 
I. -5. 4 mm. and the length of the base A C Sg mm., the coefficient 
of friclion of the ruby on the steel should be 

The angle of friction will be, in this case, 

tang. 4 ^ 0.15 
and + =; 8° 32', 

value that we will adopt in our calculations. 

C&lculKtion of the Friction In Ge&ringi. 

339. Knowing the normal pressure between the teeth of al 
gearing, one has the value of the sliding friction (333), so that if onfr I 
knows the length of the space traversed by the friction of one tooth^ 
on another, one would have the work absorbed by this friction (336). J 

Before entering into the details of this calculation, let us reca 
the kinetic question of the transmission of the movement. 

We have found that by means of gearings, the movement of one 
wheel is uniformly transmitted to another ; this geometrical demon- 
stration is independent of the material of which the wheels are formed, 
of the nature of the friction, etc. This property holds good whatever 
may be the friction in play and the greatness of the efforts which are 
shown. The passive resistances have, therefore, no eflect on the trans- 
mission of the movement, properly speaking ; they only increase J 




the work to be expended in onier to prodoce the movement of the 
motive whed. 

One thus understands that the work of fnctioa may be generally 
expressed as function ci the resisling useful work to which it is added. 
340. Let us adopt the k^owii^ notation : 

A, the whed which oontrob the movement ; 
r, its primitive radius ; 
n, its number ci teeth ; 
A\ the whed contn^led ; 
r^, its primitive radius ; 
n', its number of teeth 
a =^ t c = t d (F%. 99) the pitdi of the gearing, and Q the resis- 
tance opposed to the movement ol A acting tangendy to the primi- 
tive circumference. 



Generally, in gearings, there are several teeth of the whed 
which work at the same time ; but, in order to facilitate the calcu- 
lation, we will suppose that there is only one and that it controk 
from / to ^, that is to say, a space equal to the pitch. During this 
passage a, the work absorbed by the normal pressure between the 
teeth is equal to thb pressure multiplied by the length of the curv^e 
traversed by the point of contact / in its passage from /to /', a 
length which does not perceptibly differ from a^. But, since the 

* Let U8 note that the normal pressure has not generallf a constant ralue ; it would bare 
it onlj in the case of gearings by inTolrent of circle, if one neglected the firiction. For the 
others, it is Tariable and it is the mean Talue of this quantity that must be made to enter into 
the expression of the work of the friction. 


Lftions I 




point of contact is very slightly removed from the arete', one can 
sappOK, without committing an appreciate error, that the pressure 
Ui equal to the force Q acting tangently to / t', and the work 
absorbed is. therefore, „ g 

From the hypothesis that the normal pressure between the 
teeth ia constantly equal to Q (mean value of this pressure), i 
follows that the friction is equal to 

As to the work absorbed by this friction, one remarks that 
while the point of contact / passes to /', the space traversed by 
this point on the tooth which controls is equal to i' c and that 
which it has traversed on the controlled wheel is t' tf. from whence 
it follows that there has been sliding on a length equal to 

a difference which one can suppose equal to a straight line joiaii 
c to ir'. The work absorbed by the friction is, therefore, 

Q/X c C. 
Dropping the perpendiculars c e and c" e' to the line of centers; 
O being almost parallel to c d , one can suppose 

but, one has 

tn admit 

- and / f' = — 

c t ^ c' I = a. 

which comes nearer the truth as the pitch becomes smaller wil 
relation to the radius, one has, therefore, 

consequently the work absorbed by the friction is 

The work which the wheel A should transmit to the wheel A? 
for the distance traversed a is, therefore, 

If- - Gfl + G«4" irV + 7^) "' 

■A nhord l> the dimk iin]KirtloiuU batman the dimnuUr lud lu prajHUgn 

Gearings. 237 

If P is the motive force which acts tangently on the driving 
wheel, the motive work for the distance traversed ais P a and one has 

from whence 

34L Taking up again the formula ( i ) in which Q a represents 
the useful work '' Wu /' this can still be put under the form 

Since one has, moreover, 


n and n' being the number of teeth, and since 

n a ■= 2^ r and «' a = 2 ir r', 

one obtains ^ ^ ^ ^ 

r = and r' = 

2 IT 2 IT 

On replacing these values in the formula (2) and simplifying, 
it becomes 

Wn.= Wu ^ Wu ./» (-1- 4- -^)» (3) 


from whence one draws 

342. On examining these two last equations, we can see that 
one diminishes the friction by increasing the number of teeth. 

Thus, for a wheel of 64 teeth gearing in a pinion of 8 leaves, 
one would have, on admitting Wm = i andy= 15, 

Wu = 

+ 0.15. 3.i4i6(-^4--|-) 


while for a wheel of 96 teeth gearing in a pinion of 12 leaves, one 
would have only 


1+0.15. 3Ui6(-^ + ^) 


Lessons in Horology. 

One thus discovers the practical rule that th& number ol 

must be increased as much as possible in order to diminish the 
of friction, to have less wear and a smoother mouon. 

343. For interior gearings, the formulas (3) and {4) becomi 

^ W, + Wu 

1 +/ 



One sees that in these gearings, with the same number of teedl 
the friction is less than in the exterior gearings. 

344. The friction in the rack can also be deduced from the 
preceding formulas, on remarking that the radius of one of the 
primitive circumferences becoming infinite the general expression dt' 
the friction, for « ^ 00, becomes 

»'«=»'-+ »■» /"4-. (7) 


' + /«- 


345. For conical gearings, on preserving the same relationi 
as in the preceding cases, one would arrive at the following 
results, a. being the angle formed between the axes of the twd 

wheels : 

. - «. + »4 ^?^ ^ + ^ + '-^^ (9l 

m = - 




These gearings are smoother than cylindrical gearings of the 
same number of teeth. 

34^. Friction Before and After the Line of Centers. 

formulas which we have established, the influence of the friction is the 
same before as after the line of centers ; this result does not agree 
with those of e.vperience, which show, on the contrarj-, that the, 
friction before the line of centers is more hurtful than that which 
exerted after the passage of this line. However, since we have! 
supposed the pitch as being very small, our results can be coi 
sidered, in this case, as sufficiently exact. 

h is certain that it would no longer be the same if the contai 
commenced at a relatively great distance from the line of centers. 



347. Let us examine, for example, the case of a wheel with 60 
teeth gearing in a pinion of 6 leaves, since we know that in horology 
this gearing is one of those in which the contact of the tooth and 

the leaf should commence the most in advance of the line of centers ; 
and let us examine successively the four following cases : 

First — ^The wheel drives the pinion after the line of centers ; 

Second — ^The pinion drives the wheel before the line of centers ; 

Third — The wheel drives the pinion before the line of centers ; 

Fourth — The pinion drives the wheel after the line of centers. 

348. The Wheel Drives the Pinion After the Line of Centers. 

Suppose O and Cf the centers of two mobiles (Fig. 100), P the 

240 Lessons in Horology. 

moment of the force with relation to the axis of the wheel which 
drives, and Q the moment of the force with relation to the axis of 
the wheel which is driven. 

The wheel is in equilibrium under the action of the force P, 
whose moment is P, of the normal force N' =^ A' whose moment 
\s, ^ N. O b and of the force of frictiony^A^ directed perpendicularly 
to the normal force, and whose moment is — f N. O d. One has, 
therefore, P~N0b-fN0d=0. 

On the other hand, the wheel O is in equilibrium under the 
action of the force Q, whose moment is Q and of the normal reac- 
tion N' = A^ whose moment is — N' . & b'. The moment of the 
force of friction is null, since its lever arm is equal to zero. One 
has, therefore, 

On dividing the first of these equations by the second and 
simplifying, one obtains 


(y b' 

But, on designating by p the angle b t = V t a formed 
the normal and the line of centers, one will also have 


P r. sir 

P+/('-+ .^)cosP 


Q r'. sin P 
and, on dividing by sm p, 

g- - -5- (. + ^'^ "■"«!:?> 

or also 

^-^ = ' +/ ( I + -J) "^otang p. 

equal to 

us remark that if, 
ero, we will have 

in this equation, we make 
P r' 

the friction 

Q " •- 

an analogous formula to that which we haveestablished(i96,equa. 17). 

If, in the above formula (i), one places Q r ^ i, one will 
obtain ior P r" a value superior to unity. 

The angle a (Fig. 100) which the leaf is diverted from the line 
of centers, is the complement of p ; one can, therefore, also write 
the equation (1) : 

Gearings, 241 

Numerical Calculation. — Let Qr= i,/=o. 15, 

^ = -^. • = 42» 15' Af^ 
one has " °" 

Log: 0.165 == 0.2174839 — I 
Log: tang a = 9.9584454 

0.1759293 — I 
Number . . . = 0.14994 

We will therefore have the relation 

P r" 

-Q-^ = 1. 14994. 

On subtracting the friction, and admitting the moment P = 
I gr. , we would have, in this case, 

P_ ^ ^ 
from whence Q 6 * 

e = i.-^=o.igr. 

On introducing the force of friction, one will have 

P ^ 1.14994 X 60 
from whence Q ^ 

Q = = 0.08696 gr. 

11.4994 ^^ ^ 

349. The Pinion Drives the Wheel Before the Une of Centers. 

In this case the moment Q becomes the moving power and the 

value Q r will become superior to P r*. The formula (i) then 

becomes ^ 

Qr _ I 

I — / ^ I + — tang a J 

remarking that the sign of the friction is changed. 

Numerical Calculation. — The same data as in the pre- 
ceding case, except that we take here P r = i. We have 

/ A -f -^^ tang, a = 0.14994, 

then ' -^ (^ + ■?■) ^"^- • = ^•^5006, 

Or I , 

242 Lessons in Horology, 

On subtracting the friction, one would have, if i^ = i gr., 

-9- = A, 
from whence ^ 60 ' 

P= Q^-^ 10 gr. 

On introducing the force of friction, one will have 

= 1. 1764 

from whence ^ 60 * 

P = ^-z- = 8.5 gr. 

0.11764 ** ** 

150. The Wheel Drives the Pinion Before the Line of Centers. 

We have in this case (Fig. loi), reasoning the same as in the 

preceding cases, 

P — N. O b = O, 
Q — N'. a 1/ ^ f N\ a d = O. 

from whence 

P_ _ r. sin P . 

or, again, 8 ~ K sin. p -/ (r + r^ ) cos p ' 

Pr^ T 


1-/(1 + ^) cotang p 

But the angle b O (y, complement of p, is equal to 

a — . 

since the angles traversed in the same time by the two mobiles of a 
gearing are inversely proportional to the numbers of teeth (176). 
One will, therefore, have, 

Pr^ _ I 

Qr ~ f ( y ^\ ^ / ^\ ^^^ 


Numerical Calculation. — Let « = 17° 44' 13", j» = 60, 
»'- 6. 

One has 

i. -- = ai X 17^ 44' 15" = i^ 46' 25.3" 

/("i "^ I ) ^^ 0.15 X II = I 65 
k\^. 1.05 = 0.2174839 
lo^. rang, (i^ i?) - S.400S94S 

0,70837^7 — 2 
Number ^= 0.051095 



I — / ( I + i?) tang. ( tt i) = I — 0.051095 = 0.948905 


= 1.05384 = 


0.948905 ^^^ Q ^ 

If the moment of force /* = i gr., we have, without the 


nf 6 




\ I 

\ I 






>V ! 

Fig. 101 

On introducing the friction, one has 

Q = 

from whence 

1.05384 ■ n 1.05384 

Q = 0.09489 gr. 



Lessons in Horology. 

35L The Pinion Drives the Wheel After the line of Centers. 

The moment Q becomes the moving power and the formula (3) 
becomes n r / n \ / n'\ 

on changing the sign of the friction. 

Numerical Calculation. — One has, from the preceding^ 


-p-p ■= 1 + 0.051095 = 1.05109s. 

Without the friction, we will have 

from whence ^ * 

P = O ^-, 
andif 6= i gr., ^ '^ ^ 

With the friction, one will have 

P^^ . ' ^ TO 

6 " 1. 051095 1. 051095 

and /• = 9-514 gr- 

352. Recapitulatlan of the Preceding Calculations. The 

moment of the motive force acting on the wheel being equal to 
I gramme, the moment of the resisting force with relation to the 
axis of the pinion should be at the instant of the first contact 
before the line of centers, 

Q = 0.09489 gr.; 
and at the instant of the last contact after the line of centers, 
Q = 0.08696 gr. 
When the pinion drives the wheel, we have found at the 
instant of the first contact before the line of centers, 

/•-8,5 gr., 
and at the instant of the last contact after the line of centers, 
P = 9-514 gr- 
One sees that, in the most usual case, when the wheel drives- 
the pinion, the force absorbed by the friction before the line (A 
centers difTers very litde from that which is absorbed after the 
e of this line, which confirms what we have admitted (261),* 

r lo b« bHb lo oomparp fbeni \n sn i 

IrlresUe pinion, the mnniFDl of Ibe fa 
n is inftrior to that -'-■-'■ — >■ •— 

dleuUtcd for 

eiplsios itbj, «b«o 



One sees also that the smaller the driving wheel becomes with 
relation to the one which is driven, the more also increases the dif- 
ference of the resistance before and after the line of centers ;^e 
numbers of teeth of the two mobiles should, therefore, be ipcfeased 
[ifis much as possible. 

Colcutatiooi of the rriction of Pivott. 
353. When the watch is placed in a horizontal position, the 
E diSerent mobiles of the train rest on the flat ' ' shoulders ' ' of their 


lower pivots ; in the vertical position, these same mobiles rest on 
the cylindrical surfaces of the two pivots. The force of friction b 
proportionate to the pressure which the surfaces in contact undergo. 
In the horizontal positions, the pressure proceeds from the weight 
of the mobile and from the lateral force which presses the pivots 
against the sides of the holes ; in the vertical positions, these same 
forces are in action, but on account of the position of the gearing 
on the axes, these pressures will generally be different on each pivot. 
354. Work Absorbed by Friction on the Plane Surface of 
the Shoulder of a Pivot. Suppose (Fig. 102) r" the exterior 
radius of the shoulder and r" the radius of the pivot. Repre- 
senting also the mean radius --^ — by 8 and the width of the 
shoulder by /, then / = r" — r. 

346 Lessons in Horology. 

One will have, consequently, 

^' = 8 ^ i and r' = S — -. 
The surface of the circular crown with radius r" — r' being 

r (r"' 


one will have 

The surface of the circle with radius r' will be 

P being the pressure exerted on the crown and admitting that thia i 
pressure varies proportionately to the extent of the surface ; that I 
which is exerted on a circle with radius f' should be 



. the circle with radius r" 


and that which would take plact 
would be p / 

The work absorbed by the friction of the crown is equal to J 
the work absorbed by the friction which would be produced on the 1 
total surface of the circle with radius ^' ^ S + j- diminished by f 
that which would take place on the surface of the circle with j 
radius r" ^ s — -j' ; it is, therefore,* 

or ^, 

• Knr ■ pnuuie P. the force deieloped by rricUon it f P; Ih* " 

pradaot of//>bi the dlal»ce irnrerwd, Ttus dlsUoce li not the >Bnie for nil Ihc polnli vtM 

DbUJo lU mean •slue, divide 'thF olrcle wUb radlns r iota t number n of euui] Kclon 1 
■utBelaaUjr ■nmll ao IhU eacb of them dao be regatded u a Iriingle. The reBu^Iaol of tb« I 

([rii*llT. "T. «t K n' 'h* radius. The force of (HcHoo being for one of them / - , ' 
,^.2„^ r-ft. J»r. 
W- i fFw r. 



!•/'■(,' + r + — *7-5 — ) 

it becomes, therefore, 

k*'- ••/' (i « ^ s-j) -"""(' + As) 
3SS. In the horizontal position, the pressure P arises from the 
weight of the mobile ; this pressure is always much inferior to the 
lateral pressure with which the pivots are pressed against the sides 
of the holes. Thus, in the preceding equation, one can neglect 
the term ., 

and one has simply " 

I W = f P. 3 » 8 ==/ Pw (r" -f r*). 

B 356. In the vertical position of the watch, the pressure P on 

■■ the shoulder of the pivot is null ; one can, therefore, also admit 

that the friction is null. 

The work of the friction of the cylindrical surface of the pivots 

against the sides of the holes is expressed by 

I W = f P. ill r'. 

" r* being the radius of the pivot. The formula includes the work 
absorbed by the two pivots, since P is the total pressure and since 
the friction depends only on this pressure and not on the extent of 
the surfaces in contp.ct. 

3ST. Determination of the Lateral Pressure Received by the 
Pivots of the Mobiles in a Train. l,et us examine, for example, 
the third wheel of a watch. This mobile receives on one side an 
action on the part of the center wheel gearing in its pinion and, 
on the other, a resistance arising from the pinion of the fourth 
wheel in which the third wheel gears. These two efforts show 
themselves by a pressure on the axis, and the two pivots are pressed 
against the sides of the holes ; for each of these, the load which 
they receive can be represented in magnitude and direction by the 

24^* Lessons in Horology. 

resultant of the partial forces that the axis of the wheel receives at 
each of its extremities. This pressure depends on the relation of 
the distance of the pivots from the point of application of the forces 
in play to the length of the axis. 

358. Let us imagine the point of contact of the teeth and 
the leaves on the line of centers and let us represent by P the force 
that the leaf of the pinion receives on the part of the whed 
tooth of the center wheel ; since, simultaneously with this, one 
wheel tooth of the third wheel is pressed against a pinion leaf of the 
fourth wheel, the force P gives birth to the reaction /". The 
direction of the forces /"and P' is perpendicular to the line of centers. 

Let us call r and r" the primitive radii of the wheel and pinioo, 
we will have for the state of equilibrium 

P r' = P' r aa6 P' = P ^. 

The equilibrium will not be disturbed if one will apply to the 
point diametrically opposed to that of the contact of the tooth and 
leaf a force P^ equal, and in a contrary direction, to P and, like- 
wise, at the point opposite to the point of application of P', i 
force P\ equal and in a contrary direction to this one. 

The resultant of the forces P and P^ is 2 /* ; it should be; 
applied at the axis of the pinion, paraSlelly to the components 
(Fig. 103). 

The top pivot of the third wheel will receive on the part of the 
resultant 2 Pa force fi and the lower pivot a force q in such a w« 
that one would have ^ /» = a -I- g 

One should have p a — q b. 

a representing the distance of the shoulder of the top pivot froi 
the middle of the thickness of the center wheel, and b the distanra 
from this last point to the shoulder of the lower pivot. 

To determine the values / and q, we extract from the first 
formula q = 1 p - p. 

from whence, on substituting this value in the second, 



iK4i ^*Vv* /W^/-* ^ **m:> O B A *fXBsL a3& 1:1 «. "HTe to! isi^ a 
r^ ^f^^/^ O // A O U ff ,taABA = /. sks O.^ = 5 ^; 

/^, >/^ ^^j4^'Hfy, // ifft^i // \fy iiht haU of their Taloe, 00c obuios 

, , -r * r tf -r * 


COS a. 

I'iff IIm- luw^f |ilvo(, onr would obtain in an analogous manner 

W^i ll|Min I'HfUiilnlnM lh« Imj^. 103, one will notice that the 
pUHMMh' nt ll»i» M'nliM whrrl tcrth is greater than the resistance 
w\s\\\\ ll\»« li'i^vrftot thr» \\\\\\\ pinion oppose. Consequently, foi 


C tarings. 

the top pivot of the third wheel, the force O C ^ p will be greater 
than the force O B ^^ p' . 

For the lower pivot, the pressure oi the center wheel diminishes, 
the force O C :^ q becomes weaker, and O B" ^^ q" stronger. 
There necessarily results a different direction of the resultants R 
and ^1, which the construction of the parallelogram of the forces 

The direction of these resultants is important and enables us to 
explain the reason why one encounters, in making repairs, pivot 
holes enlarged by wear in a direction often very different from that 
where it would seem that this wear should logically be produced. 
This fact is noticed in clocks or pocket watches whose holes are not 

360. Let us note, moreover, that the problem we have just 
dealt with is based on the case of flank gearings when the con- 
tact takes place on the line of centers ; the normal forces are 
then perpendicular to the radii. But when the contact between 
the tooth and the leaf ia displaced or when the gearing is of 
I another sort, the normal forces take other directions and the 
J value of the resultants, as also their directions, can undergo a 
I slight change. 

36L The equations (i) and (a) show that, all other things 
being equal, the pressures become greatest for an angle a ^ 180°, 
for which cos a — — i ; the sign of the last term placed under the 
radical becomes then positive. From this standpoint it would, 
consequently, not be desirable to construct a train, all the mobiles of 
which would be in a straight line. 

1362. We establish also that the more the value of r increases, 
the more the pressure diminishes ; this is one of the reasons why it 
is good to increase as much as possible the diameter of the center 
and third wheels, inertia having not yet appreciable influence on 
these mobiles, 



363. Finally, it may not be useless to observe that the relation 
,- cannot be replaced by the relation of the numbers of teeth, since 

the mobiles with radii r and r" do not gear together, but are 
mounted on the same axis. 

364. Numerical Example. — Suppose 
P= 77-5 gr. fl = 0.8. b = 4.06. d = 3.2. e = 3.a. 

r' = 0.87. r = 5.49. a = 93°. 

252 Lessons in Horology, 

We will have for the top pivot 

P 77-5 ^ 0.87 

r— .- = ~^- = 15.95 • = = 0.16, 

a -^ d 4.86 ^ ^^^ r 5.49 


d* = 4.06* = 16.40. ( — ^ ) "^ (0.16 X 2.2)* = 0.12391 

2 a d — - cos a = 2 X 2.2 X 4-o6 X 0.16 X — 0.08715 = — 0.25. 

and ^ = ^5.95 1/ 16.48 + 0.1239 + 025 

^ = 15.95 1^16.8539 = 65.47. 

For the lower pivot, one will have successively 
J^i = 15-95 -Jo.S' + 0.16 X 3-2" — 2 X 0.8 X 0.32 X 0.16 X cos 95^ 

^"^ -/?! = 15.95 1/ 0.64 + 0.26 + 0.07 

^x = 15.95 1/0^97 = 15.71. 

365. Let us now determine the value of the work of friction of 

the third wheel's pivots during one oscillation of the balance. We 

have the formula (356). 

IV. F = f Pr, p. 

in which P represents the pressure, r^ the pivot's radius, p the 
angle traversed during one oscillation. 

Let us first seek this latter angle. The fourth wheel makes 
one rotation in 60 seconds or in 300 oscillations. If the third 
wheel has 75 teeth and the fourth pinion 10 leaves, this pinion 
turns 7. 5 times faster than the third wheel ; the third wheel, 
therefore, makes one turn in 300 X 7-5 oscillations = 2250 

During one oscillation it will traverse, therefore, an angle p : 

P = -^— = 0° 9' 36^^ ; 
2250 ^ ^ 

this angle expressed in length of arc with radius equal to unity is 

Qj. 0° 9^ 36-^^ •= 0.00279 

0.0028 in round numbers. 

The diameter of the pivots being 0.26, we will have for the 
top pivot 

IV. P= 0.15 X 65.47 X 0.13 X 0.0028 = 0.0035 gr. mm.. 

and for the lower pivot 

IV. P = 0.15 X 15.71 X 0.13 X 0.0028 = 0.00085 gr. nim. 

Gearings. 253 

The total work absorbed by the friction will, consequently, be 
W. F= o.exijs + 0.00085 = O.OQW5 gr. mm. 

The work of the motive force applied to this wheel being 
I 0.2 1 gr, mm. during the duration of one oscillation of the balance, 
I one can prove that the work absorbed by the friction of the pivots 
\ represents about the fiftieth part of it, 

Influence of the Oil. 
366. We have said at the beginning of the study of friction 
that the introduction of a greasy substance between the frictioning 
surfaces of two bodies, compelled to slide on each other, is neces- 
sary in all cases where a heating and consequently grinding and 
ir are to be feared. 
When greasy substances are interposed between two surfaces, 
I these are no longer in immediate contact, the molecules of grease 
liform little spheres which roll between the two bodies. In most 
I cases, especially in large mechanisms, the friction will be reduced 
I by this fact. 

In horology, especially in pocket watches, the inverse phe- 
ftnomenon can present itself. The oil which is used introduces a new 
■.resistance, an adhesion, or, otherwise expressed, a "sticking." 
■ This new resistance is added to the friction and it can happen that 
Ithe coefficient of the sura of the two resistances may be greater than 
Bthe coefficient of dry friction. With regard to the weak forces in 
I action on the last mobiles of the train, on those of the escapement 
l.and on the balance, this last resistance cannot be neglected. Unfor- 
ftunately, it is very difficult to express this force in figures, because 
■it depends on the nature of the lubricant, on its degree of fluidity 
liand on its unchangeableness. 

The friction which is exerted through the agency of a lubricant 
F depends on the speed of the bodies in contact, and on the extent of 
' their surfaces. It depends also on the nature of the movement ; 
thus, it is different on an annular balance when the latter is animated 
with a continuous circular movement and when it is animated with 
an oscillatory movement (circular reciprocating). One under- 
stands, in this latter case, that a certain quantity of oil participates 
j' in the movement of the pivots and that this oil would have a ten- 
I dency to continue in the direction of the movement, although the 
I pivots turn already in the opposite direction. 


Lessons in Horology. 

In all the experiments relative to the friction of lubricated 
bodies, care must be taken to assure oneself that the lubricants are 
neither altered nor expelled. 

One can take as a general principle that the best lubricant is 
that which is the most fluid, that is to say, it is better, when one 
can, to replace grease by oil, oil by water, water by air, which is 
equivalent to suppressing all lubricants. This supposes that the 
speed of the mobiles may be sufticiendy great not to expell the 
lubricant experimented with. But a considerable speed is necessary 
for the pieces to retain a fluid lubricant like water and with still 
much more reason for them to leave between themselves a sutfident 
cushion of air. Experiments have been made with the astonishing 
result of showing the friction almost suppressed between two pieces 
nibbing together without any lubricant and at an enormous speed.* 

This almost entire disappearance of friction is due to the inter- 
position of a cushion of air, a perfectly elastic matter, between the 
surfaces in contact. 

In horology, in all cases where the use of a lubricant is neces- 
sary, one must, therefore, take into account the speed of the 
mobiles and the pressure which they have to support. Thus, the 
wheels of the stem-winding mechanism should always be greased 
by means of a semi-fluid lubricant, f 

The motive spring, as well as the pivots of the arbor around 
which the barrel turns, should be lubricated with a thicker oil than 
that which one employs for the train and the escapement. 

The principal qualities of the refined oil which is used in 
horology should be its unchangeableness by the atmosphere and by 
the various temperatures which the watch must stand, its perfect 
flnidity and the absence of acids in its composition. The solution 
of this question so important to the preserving, for the longest pos- 
sible time, of the precision in the running of chronometrical instru- 
ments, lies within the domain of organic chemistry. 

blghiy ft 

■Dd reBDcd 


Gearings. 355 

Application of the Theory of Gearings. 

Functioni of (he Heut in Chronofiaph*. 

36?. Chronographs are horary instruments intended to measure 
very small intervals oi time. For this purpose these watches are 
furnished with a special hand fastened at the center of the dial and 
traversing a division generally exterior to the minute circle. The 
shortest interval of time measured by the hand of the chronograph 
is equal to the duration of one oscillation of the halance ; thus, 
when the balance of the watch beats 18000 oscillations per hour, the 
chronograph indicates the duration of an observation to about one- 
fifth of a second.* 

These mechanisms are of many different constructions ; their 
movement is controlled by the train of the watch, causing, by this 
fact, a slight additional burden to the motive power. Before the 
observation, the chronograph hand is fastened and remains on the 
division zero. On pressing an exterior push-piece, this hand is 
immediately put into motion ; at the end of the observation, a 
second pressure stops the hand and, finally, after reading, a third 
pressure brings it back suddenly to the division zero, where 
it remains held in place until the moment of a new use of its 

The invention of the chronograph goes as far back as the year 
1862 and is due to Adolphe Nicole, originally from the valley of 
Lake Joux but established in business in London. 

368. It does not belong to the plan of this study to give a 
description of the mechanism composing this instrument ; it will 
suffice for us to show that the action which returns the hand 
to zero on the division, is effected by the fall of a jumper on 
a heart-shaped eccentric fastened on the axis of the wheel which 
carries the chronograph hand. We will especially occupy our- 
selves here with the determination of the form to be given this 

The condition which the heart should fulfill is to present, at 
every point of its outline, a sufficient inclination to the lever (or 
jumper) which works it to assure the slipping of the lever as far as 

I the origin of the curve. 
• C. W. Scbmldt, x Svpdlih englaeer, IIiIok Id Paris, hu coostrualed chronogriplu 
Meue >ad gtop, are loteaded eipeclallT to meaniTe tlie ipeed oi prniectllei. 


lessons in Harology. 

On imagining the axis on which the heart is fixed animated 
with a condnuous circular movement and the extremity of the 
lever constantly pressed against the exterior border of the curve, 
the problem becomes, to find the form capable of changing, 
uniformly, a continuous circular movement into a reciprocating ■ 
circular movement. 

369. Lei us first examine the simpler case of the transforma- 
tion of a continuous circular movement into reciprocating rectilinear 
movement, by means of a heart-shaped eccentric. 

Since the reciprocating movement should be uniform, the 
point B of the line A B (Fig. 104) should successively occupy the J 

equidistant positions B, i, 2, 3, 4, -i, the lengths B i, 12, 2j, 

^, . . . . being supposed equal fracUonal parts of the total j 
course B A. If, from the point O as center, one described cir- 
cumferences passing through the points B, i, z, 3, ^ A, 

and if one divides the circumference whose radius \s O A into I 
the same number of divisions into which the line A B has ( 
been divided, the intersections of the circumierences, with the 
radii passing through the points of division, will indicate succes- 
sively the points through which should be described the envelope 
curve of the point B. By construction, the uniform movement 



! Horology. 

of this curve around the center O will communicate a unifomt 
rectihnear movement to the point B, alternately from AtoB 
from B to A, tlie lower part of the curve naturally being symmet- 
rical with the upper part. One recognizes thus that the form 
obtained is that of a spiral of Archimedes, elongated, whose equa- 
tion is 

% = aw^ C: 

the radius vector S less a constant quantity, C, is always propor- 
tional to the angle described, w. 

370. In chronographs, the reciprocating movement of the 
point B is rot rectilinear ; it is circular and its movement ift 
executed around a center C (Fig. 105). 

Let us, therefore, now determine the shape of the heart 
suitable for the above new condition, and let us make the 
pointed end of a lever (7 B traverse an arc A B with a uniform 
movement, while the axis O turns the arc B D C^ ^ r^. Let' 
us divide the arcs A B and B D C into the same number of 
equal parts, and let us describe from the center O concentric 
circumferences passing through the division points of the arc A B 
draw afterward the radii passing through the division points of the 
3XK. B D C. 

In order to determine, now, the points m, n, o. A' and' 
m' , n', o', of the curve of the heart, let us consider, as in the 
preceding case, the intersections of the circumferences and of the 
radii, but let us lay off here in addition the lengths of the 
arcs included on each circumference between the initial radius 
B E and the arc descrit>ed by the point B of the lever, before 
the points of intersection considered. One would thus obtain 
the curve B ?n n o A' o' n' m\ which would fulfill the con- 
ditions desired. In fact, supposing, for example, the point o of 
the heart arrived at the point j of the arc B A. the radius 
K will be superposed on the radius B E, the axis of the 
heart has completed three-quarters of its course and in the same 
time the point B of the lever will have been lifted up three- 
quarters of the distance B A. 

371. Let us decide to determine by calculation the value 
of the radius vector of the heart corresponding to the point 
of the lever for any position whatever of the axis of the eccen- 
tric (Fig- 106). 




Fig. 106 

3 Lessons in Horology. 

Suppose ; 
, ihe distance between the center of rotation of the heart and 

that of the arm (^R should also be the distance from the point 

of contact to the center of the arm) ; 
, the variable radius vector from the center of the heart to any 

point whatever of the exterior curve ; 
, the radius vector of the heart corresponding to the position 

repose, when the hand of the chronograph is at zero ; 
, the greatest radius vector of the heart ; 
, the angle formed by the radii vector r^ and r ; 


of centers O (7 and the radius R 
contact with the radius vector r 

contact with the radius r 
contact with the radius r 


B. the angle formed by the li 

of the arm when the latter 

of the heart ; 

e„, this angle when the arm is 

e', this angle when the arm is 

The radius r is, therefore, the chord of 
radius R and corresponding to angle e ; one 
r = 2 V?. sin i 0. 

The angle a, formed by the radius vector r^ and the radius 
considered r, is different from the angle which the heart should 
turn starling from the position of repose to the instant when the 
radius r coincides with the point of the lever. Designating thufl 
last angle by -y, we have in effect 

T = » 
according as the axis turns to the left or to the right, p being t 
angle formed by the fixed direction of the radius r^ and that whicj 
the radius r lakes at the moment of contact with the lever. Let us^ 
P'ace p ^ J (fl _ fl^), 

this angle being inscribed in the circumference with radius j 
and e — flj angle at the center embracing the same arc. We w3t J 
have, consequendy, ^^ ^ + (e — e„). 

Let us now establish the relation which connects the angU^J 
e — e„ and -y, angles which should be in a determined relation, 1 
virtue of the mechanical principle stating that the transmission e 
the force is uniform when the angles traversed in the same time b 
two mobiles which drive each other remain constantly ii 
relation. But when the lever traverses the total angle 8' — iJ 

Gearings, 261 

the heart executes a demi-turn, therefore, an angle equal to v ; one 
will, therefore, have ^ ^ 

from whence 


Consequently, one can write 

e — e 

and ^' "~ ^^ 

e^ -K 

a = '»r ^,---V^i(e^0^) 

and also w Oo '^ »o 

from whence e 

a 4- w e^zre"" =^ i ^o = » j73:t" "^ * *» 

One will thus have 


The equation of the two branches of the curve of the heart 
expressed in polar co-ordinates will, therefore, be 

r = 2 i?. sin i (—^ -h eo^ 

372. Numerical Calculation. — Let us admit 

r^ = 2 R. sin J 6^ = 4. r^= 2 R. sin J 6-' ^ 24. 7? = 140. 

We will have . . ^ 4 , . , a, 24 

sin 4 80 = -~ and sin 4 8^= -~, 
Which gives ^280 ^280 

•o = 1° 38' ^Z^' 
consequently, •' = 9" 5c/ a^'' ; 

8^ — 80 = 8° 11^ 49.4^^ 

Expressed in seconds of the arc, the angles 8^, 8^ — 8^ and v give 

®o = 5893.6 seconds 
6^ — 60 = 29509.4... ** 

IT = 684000 . . . ** 

Let us first calculate this equation under the form 
r = 2 
and suppose a = 30®. 

^ sin i (^,qr xfe^^^ej + •-^ 

Lessons in Horology. 

i(e'-«<,) = 

■ log: 

648000" log 


The angle a - 
log : 108000 

+ i < y - fl. ) 

t«+ i (»'-fl.)) =5-8213528 

— log : (8' - fl) = 4.4699589 

1. 35 1 3939 

: 30° ^ 108000" 
= 50334238 

4808.72 ■ 


3.6820299 log: 4808.72". 
i.72 -i- 5893-6 ^ 10702.32"; 

from whence 

log : 2 .^ sin I B 

r = 7.36327 for > 


Similar calculations will give successively 

r = 10.5254 for a = 60 
r = 13.786:; ■' a = 90 

r = 17045' " a = 120 

r = 20.3017 ■■ B = 15c 
r = 23.5623 ■' o == i8c 

For the other branch of the 
r ^ 2 Jf s\a\ 

s the formula would be 


and identical calculations to the preceding would give the follow 
results : 

r = 7-4153 'or B 

r = 10.8294 

r ^ 14.240 

;- = 17.66 

>- = 21.06 

r = 24.4648 

For this last calculation, the radius r no longer belongs to the 
closed curve, but rather to the prolongation of this curve. Let us 
remark that the greatest radius vector should be equal to 24. 



If one would like to know, also, the value of the angle p, cor- 
responding to the above data, one would have 

373. In order to obtain a greater stability for 
the chronograph hand, one prefers, sometimes, to 
make the heart like the adjoining form (Fig. 107) ; 
one suppresses, in doing this, a part of its curve, 
but, by way of compensation, the part B A B' oi 
the arm is terminated by curves fulfilling the 
conditions desired. 

Fli;. 107 

Note.— In the original French, paragraphs 374 to 377, indn* 
sive, appeared at the end of Vol. I as an appendix, but in trans- 
lating were moved to their proper places in the text and the 
pan^praph numbers, together with the figure numbers under 
cuts, were accordingly changed as follows : 

Paragraph No. 374 now appears as No. 169 a on page 114 
Paragraph No. 875 now appears as No. 169 h on page 115 
Paragraph No. 876 now appears as No. 169 c on page 115 
Paragraph No. 377 now appears as No. 226 a on page 154 
Paragraph No. 378 now appears as No. 226 h on page 155 
Figure No. 108 now appears as No. 37 a on page 114 
Figoie No. 109 now appears as No. 61 a on sage 156 



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VOL. I. 

Bv C. H. Brown. M. D. 

ebJwitiaBia PkiteMpfaU H< 

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A study o( it is essential to an intelligent 
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Chapter I.— Introductory Remarks. 

Chapter II.— The Eye Anatomically. 

Chapter III.— The Eye Optically; or. The Physiology of Vision. 

Chapter IV.— Optics. 

Chipter v.— Lensea. 

Chapter VI.— Numbering of Lenses. 

Chapter VII.— The Use and Value of Qlasses. 

Chapter VII I.— Outfit Required. 

Chapter IX.-~Method of Examination. 

Chapter X.— Presbyopia. 

The Optician's Manual, Vol. I., is complete in itself, and 
han been the entire optical education of many successful opti- 
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being simple in style, accurate in statement and comprehensive 
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Bv CarlWeiland, M.D. 

Eliilo3tlphta,''pa. ' ' '™"' 

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in the Human Eye, Based on the Laws of 
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Bi Charles F. Prentice, M. C. 

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This book of over 200 pages contains the following papers : 

The PriBm-Dlopi 

tar Comlriaed CyliadHcffl Lemeit 
il Nuraberfais and Measorlnfc Prim 

The PrUm 

On tile Practical Etecullon ol Opbthallnic PreKriptlons invDlvlns Prinni. 
A Problem Id Cemented Bifocal Lenaei. Solved by tbe Pri^m-Dioplry. 
Why Stmne Conlra-aeneric Lense* of Equal Power Pall to Neutralize 

Correction of Depleted DyDamlc Refraction (Presbyopia) 

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